can any of u do the 6 questions attached in the hw -3 pdf file , i am  attaching book as well as lecture notes , i need it done in 20 hrs ,  please reply asap  

MATH57

2

Spring 2020
Assignment 3

Due: Wednesday March

1

1
Solve any set of problems for 100 points.

Problem 1: (30 points) Let Ω ⊂ R2 and u ∈ H1(Ω). Prove the following inequality∫
∂

Ω

u2dx ≤ C‖u‖21 (1)∫
Ω

u2dx ≤ C
[∫

Ω

|∇u|2dx +
∫
∂Ω

u2dx

]
(2)

Problem 2: (30 points) Consider the following B.V.P for elliptic equation in R2.

−∆u + q(x)u = f(x,y), (x,y) ∈ Ω = (0, 2) × (0, 1)

and
∂u

∂ν
= g, (x,y) ∈ Γ,

where Γ is its boundary, and ν is the outward unit normal vector to Γ. Here q = 1 in (0, 1)×(0, 1)
and q = 0 in the remaining part of the domain.
Derive the weak formulation for this problem and show the coercivety of the corresponding bilin-
ear form in H(Ω)− norm.

Problem 3: (50 points) Consider the τ to be a tetrahedron in the (x,y,x)− space determined
by its vertexes P1,P2,P3,P4. In order these four points to form a tetrahedron we assume that they
are not in a plane . Let Σ = {v(P1),v(P2),v(P3),v(P4)} be the set of values of a function v at
the vertices. Find a nodal basis for the space of linear functions over τ by using homogeneous
(baracentric) coordinates (λ1,λ2,λ3,λ4). Compute the element mass matrix.

Problem 4: (20 points) Let Ω be the square (0, 1) × (0, 1). Prove the Poincare inequality

‖u‖2L2(Ω) ≤ C

(
‖∇u‖2L2(Ω) +

(∫
Ω

u dx

)2

)
.

Problem 5: (20 points) Let τ be a shape regular square in 2 − D with a side hτ . If ∂τ is the

1

boundary of τ show that there is a constant C independent of hτ such that

‖v‖2L2(∂τ) ≤ C
(
h−1τ ‖v‖

2
L2(τ) + hτ‖∇v‖

2
L2(τ)

)
.

Problem 6: (20 points) Consider (τ,P, Σ), where

τ = {rectangle (xi−1,xi) × (yj−1,yj) with vertices P1,P2,P3,P4};

P = {v : v(x,y) = a00 + a10x + a01y + a11xy + a20x2 + a21x2y + a12xy2 + a02y2};

Σ = {v(P1),v(P2),v(P3),v(P4),v(P12),v(P23),v(P34),v(P41)}

where Pij is the mid point of the edge joining Pi and Pj. Show that the set Σ is P− unisolvent.
Note that the term x2y2 is missing in the polynomial set and the center of the rectangle is allso
missing from the set of points values so that dimP = 8 and the number of degrees of freedom is 8.

2

Texts in Applied Mathematics 45
Editors
J.E. Marsden
L. Sirovich
S.S. Antman
Advisors
G. Iooss
P. Holmes
D. Barkley
M. Dellnitz
P. Newton

Texts in Applied Mathematics
1. Sirovich: Introduction to Applied Mathematics.
2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos.
3. Hale/Koçak: Dynamics and Bifurcations.
4. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, Third Edition.
5. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary
Differential Equations.
6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems,
Second Edition.
7. Perko: Differential Equations and Dynamical Systems, Third Edition.
8. Seaborn: Hypergeometric Functions and Their Applications.
9. Pipkin: A Course on Integral Equations.
10. Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences,
Second Edition.
11. Braun: Differential Equations and Their Applications, Fourth Edition.
12. Stoer/Bulirsch: Introduction to Numerical Analysis, Third Edition.
13. Renardy/Rogers: An Introduction to Partial Differential Equations.
14. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and
Applications.
15. Brenner/Scott: The Mathematical Theory of Finite Element Methods, Second Edition.
16. Van de Velde: Concurrent Scientific Computing.
17. Marsden/Ratiu: Introduction to Mechanics and Symmetry, Second Edition.
18. Hubbard/West: Differential Equations: A Dynamical Systems Approach:
Higher-Dimensional Systems.
19. Kaplan/Glass: Understanding Nonlinear Dynamics.
20. Holmes: Introduction to Perturbation Methods.
21. Curtain/Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory.
22. Thomas: Numerical Partial Differential Equations: Finite Difference Methods.
23. Taylor: Partial Differential Equations: Basic Theory.
24. Merkin: Introduction to the Theory of Stability.
25. Naber: Topology, Geometry, and Gauge Fields: Foundations.
26. Polderman/Willems: Introduction to Mathematical Systems Theory: A Behavioral
Approach.
27. Reddy: Introductory Functional Analysis: with Applications to Boundary Value
Problems and Finite Elements.
28. Gustafson/Wilcox: Analytical and Computational Methods of Advanced Engineering
Mathematics.
29. Tveito/Winther: Introduction to Partial Differential Equations: A Computational
Approach.
30. Gasquet/Witomski: Fourier Analysis and Applications: Filtering, Numerical
Computation, Wavelets.
31. Brémaud: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues.
32. Durran: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics.
33. Thomas: Numerical Partial Differential Equations: Conservation Laws and Elliptic
Equations.
(continued after index)

Stig Larsson · Vidar Thomée
Partial
Differential Equations
with Numerical
Methods
123

Stig Larsson
Vidar Thomée
Mathematical Sciences
Chalmers University of Technology
and University of Gothenburg
412 96 Göteborg
Sweden
stig@chalmers.se
thomee@chalmers.se
Series Editors
J.E. Marsden
Control and Dynamical Systems, 107-81
California Institute of Technology
Pasadena, CA 91125
USA
marsden@cds.caltech.edu
L. Sirovich
Laboratory of Applied Mathematics
Mt. Sinai School of Medicine
Box 1012
New York City, NY 10029-6574
USA
lawrence.sirovich@mssm.edu
S.S. Antman
Department of Mathematics
and
Institute for Physical Science
and Technology
University of Maryland
College Park, MD 20742-4015
USA
ssa@math.umd.edu
First softcover printing 2009
ISBN 978-3-540-88705-8 e-ISBN 978-3-540-88706-5
DOI 10.1007/978-3-540-88706-5
Texts in Applied Mathematics ISSN 0939-2475
Library of Congress Control Number: 2008940064
Mathematics Subject Classification (2000): 35-01, 65-01
c⃝ 2009, 2003 Springer-Verlag Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from Springer. Violations
are liable to prosecution under the German Copyright Law.
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
even in the absence of a specific statement, that such names are exempt from the relevant protective laws
and regulations and therefore free for general use.
Coverdesign: WMXDesign GmbH, Heidelberg
Printed on acid-free paper
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Series Preface
Mathematics is playing an ever more important role in the physical and
biological sciences, provoking a blurring of boundaries between scientific
disciplines and a resurgence of interest in the modern as well as the classical
techniques of applied mathematics. This renewal of interest, both in re-
search and teaching, has led to the establishment of the series Texts in
Applied Mathematics (TAM).
The development of new courses is a natural consequence of a high level
of excitement on the research frontier as newer techniques, such as numeri-
cal and symbolic computer systems, dynamical systems, and chaos, mix
with and reinforce the traditional methods of applied mathematics. Thus,
the purpose of this textbook series is to meet the current and future needs
of these advances and to encourage the teaching of new courses.
TAM will publish textbooks suitable for use in advanced undergraduate
and beginning graduate courses, and will complement the Applied Mathe-
matical Sciences (AMS) series, which will focus on advanced textbooks and
research-level monographs.
Pasadena, California J.E. Marsden
New York, New York L. Sirovich
College Park, Maryland S.S. Antman

Preface
Our purpose in this book is to give an elementary, relatively short, and hope-
fully readable account of the basic types of linear partial differential equations
and their properties, together with the most commonly used methods for their
numerical solution. Our approach is to integrate the mathematical analysis
of the differential equations with the corresponding numerical analysis. For
the mathematician interested in partial differential equations or the person
using such equations in the modelling of physical problems, it is important
to realize that numerical methods are normally needed to find actual values
of the solutions, and for the numerical analyst it is essential to be aware that
numerical methods can only be designed, analyzed, and understood with suf-
ficient knowledge of the theory of the differential equations, using discrete
analogues of properties of these.
In our presentation we study the three major types of linear partial differ-
ential equations, namely elliptic, parabolic, and hyperbolic equations, and for
each of these types of equations the text contains three chapters. In the first
of these we introduce basic mathematical properties of the differential equa-
tion, and discuss existence, uniqueness, stability, and regularity of solutions
of the various boundary value problems, and the remaining two chapters are
devoted to the most important and widely used classes of numerical methods,
namely finite difference methods and finite element methods.
Historically, finite difference methods were the first to be developed and
applied. These are normally defined by looking for an approximate solution
on a uniform mesh of points and by replacing the derivatives in the differential
equation by difference quotients at the mesh-points. Finite element methods
are based instead on variational formulations of the differential equations and
determine approximate solutions that are piecewise polynomials on some par-
tition of the domain under consideration. The former method is somewhat
restricted by the difficulty of adapting the mesh to a general domain whereas
the latter is more naturally suited for a general geometry. Finite element
methods have become most popular for elliptic and also for parabolic prob-
lems, whereas for hyperbolic equations the finite difference method continues
to dominate. In spite of the somewhat different philosophy underlying the
two classes it is more reasonable in our view to consider the latter as further

Preface
developments of the former rather than as competitors, and we feel that the
practitioner of differential equations should be familiar with both.
To make the presentation more easily accessible, the elliptic chapters are
preceded by a chapter about the two-point boundary value problem for a
second order ordinary differential equation, and those on parabolic and hy-
perbolic evolution equations by a short chapter about the initial value prob-
lem for a system of ordinary differential equations. We also include a chapter
about eigenvalue problems and eigenfunction expansion, which is an impor-
tant tool in the analysis of partial differential equations. There we also give
some simple examples of numerical solution of eigenvalue problems.
The last chapter provides a short survey of other classes of numerical
methods of importance, namely collocation methods, finite volume methods,
spectral methods, and boundary element methods.
The presentation does not presume a deep knowledge of mathematical and
functional analysis. In an appendix we collect some of the basic material that
we need in these areas, mostly without proofs, such as elements of abstract
linear spaces and function spaces, in particular Sobolev spaces, together with
basic facts about Fourier transforms. In the implementation of numerical
methods it will normally be necessary to solve large systems of linear algebraic
equations, and these generally have to be solved by iterative methods. In a
second appendix we therefore include an orientation about such methods.
Our purpose has thus been to cover a rather wide variety of topics, notions,
and ideas, rather than to expound on the most general and far-reaching
results or to go deeply into any one type of application. In the problem
sections, which end the various chapters, we sometimes ask the reader to
prove some results which are only stated in the text, and also to further
develop some of the ideas presented. In some problems we propose testing
some of the numerical methods on the computer, assuming that Matlab or
some similar software is available. At the end of the book we list a number
of standard references where more material and more detail can be found,
including issues concerned with implementation of the numerical methods.
This book has developed from courses that we have given over a rather
long period of time at Chalmers University of Technology and Göteborg Uni-
versity originally for third year engineering students but later also in begin-
ning graduate courses for applied mathematics students. We would like to
thank the many students in these courses for the opportunities for us to test
our ideas.
Göteborg, Stig Larsson
January, 2003 Vidar Thomée
In the second printing 2005 we have corrected several misprints and minor
inadequacies, and added a few problems. SL & VT
VIII

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation and Mathematical Preliminaries . . . . . . . . . . . . . . . . . . 4
1.3 Physical Derivation of the Heat Equation . . . . . . . . . . . . . . . . . . 7
1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 A Two-Point Boundary Value Problem . . . . . . . . . . . . . . . . . . . 15
2.1 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 A Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Dirichlet’s Problem for a Disc. Poisson’s Integral . . . . . . . . . . . 28
3.4 Fundamental Solutions. Green’s Function . . . . . . . . . . . . . . . . . . 30
3.5 Variational Formulation of the Dirichlet Problem . . . . . . . . . . . 32
3.6 A Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Finite Difference Methods for Elliptic Equations . . . . . . . . . . 43
4.1 A Two-Point Boundary Value Problem . . . . . . . . . . . . . . . . . . . . 43
4.2 Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Finite Element Methods for Elliptic Equations . . . . . . . . . . . . 51
5.1 A Two-Point Boundary Value Problem . . . . . . . . . . . . . . . . . . . . 51
5.2 A Model Problem in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Some Facts from Approximation Theory . . . . . . . . . . . . . . . . . . . 60
5.4 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.5 An A Posteriori Error Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.7 A Mixed Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Contents
6 The Elliptic Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.1 Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Numerical Solution of the Eigenvalue Problem . . . . . . . . . . . . . 88
6.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Initial-Value Problems for ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.1 The Initial Value Problem for a Linear System . . . . . . . . . . . . . 95
7.2 Numerical Solution of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.1 The Pure Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Solution by Eigenfunction Expansion . . . . . . . . . . . . . . . . . . . . . 114
8.3 Variational Formulation. Energy Estimates . . . . . . . . . . . . . . . . 120
8.4 A Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9 Finite Difference Methods for Parabolic Problems . . . . . . . . 129
9.1 The Pure Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.2 The Mixed Initial-Boundary Value Problem . . . . . . . . . . . . . . . . 138
9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10 The Finite Element Method for a Parabolic Problem . . . . . 149
10.1 The Semidiscrete Galerkin Finite Element Method . . . . . . . . . . 149
10.2 Some Completely Discrete Schemes . . . . . . . . . . . . . . . . . . . . . . . 156
10.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
11 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
11.1 Characteristic Directions and Surfaces . . . . . . . . . . . . . . . . . . . . 163
11.2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
11.3 First Order Scalar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
11.4 Symmetric Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
12 Finite Difference Methods for Hyperbolic Equations . . . . . . 185
12.1 First Order Scalar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.2 Symmetric Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
12.3 The Wendroff Box Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
13 The Finite Element Method for Hyperbolic Equations . . . . 201
13.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
13.2 First Order Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . 205
13.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
X

Contents
14 Some Other Classes of Numerical Methods . . . . . . . . . . . . . . . 217
14.1 Collocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
14.2 Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
14.3 Finite Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
14.4 Boundary Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
14.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
A Some Tools from Mathematical Analysis . . . . . . . . . . . . . . . . . . 225
A.1 Abstract Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
A.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A.3 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
B Orientation on Numerical Linear Algebra . . . . . . . . . . . . . . . . . 245
B.1 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
B.2 Iterative Methods. Relaxation, Overrelaxation,
and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
B.3 Alternating Direction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
B.4 Preconditioned Conjugate Gradient Methods . . . . . . . . . . . . . . . 249
B.5 Multigrid and Domain Decomposition Methods . . . . . . . . . . . . 250
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
XI

1 Introduction
In this first chapter we begin in Sect. 1.1 by introducing the partial differ-
ential equations and associated initial and boundary value problems that we
shall study in the following chapters. The equations are classified into ellip-
tic, parabolic, and hyperbolic equations, and we indicate the corresponding
type of problems in physics that they model. We discuss briefly the concept
of a well posed boundary value problem, and the various techniques used in
our subsequent presentation. In Sect. 1.2 we introduce some notation and
concepts that will be used throughout the text, and in Sect. 1.3 we include a
detailed derivation of the heat equation from physical principles explaining
the meaning of all terms that occur in the equation and the boundary con-
ditions. In the problem section, Sect. 1.4, we add some further illustrative
material.
1.1 Background
In this text we study boundary value and initial-boundary value problems for
partial differential equations, that are significant in applications, from both
a theoretical and a numerical point of view. As a typical example of such a
boundary value problem we consider first Dirichlet’s problem for Poisson’s
equation,
−∆u = f (x) in Ω,(1.1)
u = g(x) on Γ,(1.2)
where x = (x1, . . . , xd), ∆ is the Laplacian defined by ∆u =
∑d
j=1 ∂
2u/∂x2j ,
and Ω is a bounded domain in d-dimensional Euclidean space Rd with bound-
ary Γ . The given functions f = f (x) and g = g(x) are the data of the problem.
Instead of Dirichlet’s boundary condition (1.2) one can consider, for instance,
Neumann’s boundary condition
(1.3)
∂u
∂n
= g(x) on Γ,
where ∂u/∂n denotes the derivative in the direction of the exterior unit nor-
mal n to Γ . Another choice is Robin’s boundary condition

2 1 Introduction
(1.4)
∂u
∂n
+ β(x)u = g(x) on Γ.
More generally, a linear second order elliptic equation is of the form
(1.5) Au := −
d∑
i,j=1
∂
∂xi
(
aij (x)
∂u
∂xj
)
+
d∑
j=1
bj (x)
∂u
∂xj
+ c(x)u = f (x),
where A(x) = (aij (x)) is a sufficiently smooth positive definite matrix, and
such an equation may also be considered in Ω together with various bound-
ary conditions. In our treatment below we shall often restrict ourselves, for
simplicity, to the isotropic case A(x) = a(x)I, where a(x) is a smooth positive
function and I the identity matrix.
Elliptic equations such as the above occur in a variety of applications,
modeling, for instance, various potential fields (gravitational, electrostatic,
magnetostatic, etc.), probability densities in random-walk problems, station-
ary heat flow, and biological phenomena. They are also related to important
areas within pure mathematics, such as the theory of functions of a com-
plex variable z = x + iy, conformal mapping, etc. In applications they often
describe stationary, or time independent, physical states.
We also consider time dependent problems, and our two model equations
are the heat equation,
(1.6)
∂u
∂t
− ∆u = f (x, t),
and the wave equation,
(1.7)
∂2u
∂t2
− ∆u = f (x, t).
These will be considered for positive time t, and for x varying either through-
out Rd or in some bounded domain Ω ⊂ Rd, on the boundary of which
boundary conditions are prescribed as for Poisson’s equation above. For these
time dependent problems, the value of the solution u has to be given at the
initial time t = 0, and in the case of the wave equation, also the value of
∂u/∂t at t = 0. In the case of the unrestricted space Rd the respective prob-
lems are referred to as the pure initial value problem or Cauchy problem and,
in the case of a bounded domain Ω, a mixed initial-boundary value problem.
Again, these equations, and their generalizations permitting more general
elliptic operators than the Laplacian ∆, appear in a variety of applied con-
texts, such as, in the case of the heat equation, in the conduction of heat in
solids, in mass transport by diffusion, in diffusion of vortices in viscous fluid
flow, in telegraphic transmission in cables, in the theory of electromagnetic
waves, in hydromagnetics, in stochastic and biological processes; and, in the
case of the wave equation, in vibration problems in solids, in sound waves in

1.1 Background 3
a tube, in the transmission of electricity along an insulated, low resistance
cable, in long water waves in a straight canal, etc.
Some characteristics of equations of type (1.7) are shared with certain
systems of first order partial differential equations. We shall therefore also
have reason to study scalar linear partial differential equations of the form
∂u
∂t
+
d∑
j=1
aj (x, t)
∂u
∂xj
+ a0(x, t)u = f (x, t),
and corresponding systems where the coefficients are matrices. Such systems
appear, for instance, in fluid dynamics and electromagnetic field theory.
Applied problems often lead to partial differential equations which are
nonlinear. The treatment of such equations is beyond the scope of this pre-
sentation. In many cases, however, it is useful to study linearized versions of
these, and the theory of linear equations is therefore relevant also to nonlinear
problems.
In applications, the equations used in the models normally contain physi-
cal parameters. For instance, in the case of the heat conduction problem, the
temperature at a point of a homogeneous isotropic solid, extended over Ω,
with the thermal conductivity k, density ρ, and specific heat capacity c, and
with a heat source f (x, t), satisfies
ρ c
∂u
∂t
= ∇ · (k∇u) + f (x, t) in Ω.
If ρ, c, and k are constant, this equation may be written in the form (1.6)
after a simple transformation, but if they vary with x, a more general elliptic
operator is involved.
In Sect. 1.3 below we derive the heat equation from physical principles and
explain, in the context given, the physical meaning of all terms in the elliptic
operator (1.5) as well as the boundary conditions (1.2), (1.3), and (1.4).
A corresponding derivation or the wave equation is given in Problem 1.2.
Boundary value problems for elliptic equations, or stationary problems, may
appear as limiting cases of the evolution problems as t → ∞.
One characteristic of mathematical modeling is that once the model is
established, in our case as an initial or initial-boundary value problem for a
partial differential equation, the analysis becomes purely mathematical and is
independent of any specific application that the model describes. The results
obtained are then valid for all the different examples of the model. We shall
therefore not use much terminology from physics or other applied fields in our
exposition, but invoke special applications in the exercises. It is often conve-
nient to keep such examples in mind to enhance the intuitive understanding
of a mathematical model.
The equations (1.1), (1.6), and (1.7) are said to be of elliptic, parabolic,
and hyperbolic type, respectively. We shall return to the classification of

4 1 Introduction
partial differential equations into different types in Chapt. 11 below, and
note here only that a differential equation in two variables x and t of the
form
a
∂2u
∂t2
+ 2b
∂2u
∂x∂t
+ c
∂2u
∂x2
+ . . . = f (x, t)
is said to be elliptic, hyperbolic or parabolic depending on whether δ = ac−b2
is positive, negative, or zero. Here . . . stands for a linear combination of
derivatives of orders at most 1. In particular,
∂2u
∂t2
+
∂2u
∂x2
= f (x, t),
∂2u
∂t2
−
∂2u
∂x2
= f (x, t),
and
∂u
∂t
−
∂2u
∂x2
= f (x, t),
are of these three types, respectively. Note that the conditions on the sign
of δ are the same as those occurring in the classification of plane quadratic
curves into ellipses, hyperbolas, and parabolas.
Together with the partial differential equations we also study numerical
approximations by finite difference and finite element methods. For these
problems, the continuous and the discretized equations, we prove results of
the following types:
– existence of solutions,
– uniqueness of solutions,
– stability, or continuous dependence of solutions with respect to perturba-
tions of data,
– error estimates (for numerical methods).
A boundary value problem that satisfies the three first of these conditions
is said to be well posed. In order to prove such results we employ several
techniques:
– maximum principles,
– Fourier methods; these are techniques that are based on the use of the
Fourier transform, Fourier series expansion, or eigenfunction expansion,
– energy estimates,
– representation of solution operators by means of Green’s functions.
1.2 Notation and Mathematical Preliminaries
In this section we briefly introduce some basic notation that will be used
throughout the book. For more details on function spaces and norms we refer
to App. A.

1.2 Notation and Mathematical Preliminaries 5
By R and C we denote the sets of real and complex numbers, respectively,
and we write
Rd =
{
x = (x1, . . . , xd) : xi ∈ R, i = 1, . . . , d
}
, R+ =
{
t ∈ R : t > 0
}
.
A subset of Rd is called a domain if it is open and connected. By Ω we
usually denote a bounded domain in Rd, for i = 1, 2, or 3 (if d = 1, then Ω is
a bounded open interval). Its boundary ∂Ω is usually denoted Γ . We assume
throughout that Γ is either smooth or a polygon (if d = 2) or polyhedron (if
d = 3). By Ω̄ we denote the closure of Ω, i.e., Ω̄ = Ω∪Γ . The (length, area, or)
volume of Ω is denoted by |Ω|, the volume element in Rd is dx = dx1 · · · dxd,
and ds denotes the element of arclength (if d = 2) or surface area (if d = 3)
on Γ . For vectors in Rd we use the Euclidean inner product x·y =
∑d
i=1 xiyi
and norm |x| =
√
x · x.
Let u, v be scalar functions and w = (w1, . . . , wd) a vector-valued function
of x ∈ Rd. We define the gradient, the divergence, and the Laplace operator
(Laplacian) by
∇v = grad v =
( ∂v
∂x1
, . . . ,
∂v
∂xd
)
,
∇ · w = div w =
d∑
i=1
∂wi
∂xi
,
∆v = ∇ · ∇v =
d∑
i=1
∂2v
∂x2i
.
We recall the divergence theorem
∫
Ω
∇ · w dx =
∫
Γ
w · n ds,
where n = (n1, . . . , nd) is the outward unit normal to Γ . Applying this to the
product wv we obtain Green’s formula:
∫
Ω
w · ∇v dx =
∫
Γ
w · n v ds −
∫
Ω
∇ · w v dx.
When applied with w = ∇u the formula becomes
∫
Ω
∇u · ∇v dx =
∫
Γ
∂u
∂n
v ds −
∫
Ω
∆u v dx,
where ∂u/∂n = n · ∇u is the exterior normal derivative of u on Γ .
A multi-index α = (α1, . . . , αd) is a d-vector where the αi are non-negative
integers. The length |α| of a multi-index α is defined by |α| =
∑d
i=1 αi. Given
a function v : Rd → R we may write its partial derivatives of order |α| as

6 1 Introduction
(1.8) Dαv =
∂|α|v
∂xα11 · · · ∂x
αd
d
.
A linear partial differential equation of order k in Ω can therefore be written
∑
|α|≤k
aα(x)D
αu = f (x),
where the coefficients aα(x) are functions of x in Ω. We also use subscripts
to denote partial derivatives, e.g.,
vt = Dtv =
∂v
∂t
, vxx = D
2
xv =
∂2v
∂x2
.
For M ⊂ Rd we denote by C(M ) the linear space of continuous functions
on M , and for bounded continuous functions we define the maximum-norm
(1.9) ∥v∥C(M) = sup
x∈M
|v(x)|.
For example, this defines ∥v∥C(Rd). When M is a bounded and closed set,
i.e., a compact set, the supremum in (1.9) is attained and we may write
∥v∥C(M ) = max
x∈M
|v(x)|.
For a not necessarily bounded domain Ω and k a non-negative integer we
denote by Ck(Ω) the set of k times continuously differentiable functions in Ω.
For a bounded domain Ω we write Ck(Ω̄) for the functions v ∈ Ck(Ω) such
that Dαv ∈ C(Ω̄) for all |α| ≤ k. For functions in Ck(Ω̄) we use the norm
∥v∥Ck(Ω̄) = max
|α|≤k
∥Dαv∥C(Ω̄),
and the seminorm, including only the derivatives of highest order,
|v|Ck(Ω̄) = max
|α|=k
∥Dαv∥C(Ω̄).
When we are working on a fixed domain Ω we often omit the set in the
notation and write simply ∥v∥C, |v|Ck , etc.
By Ck0 (Ω) we denote the set of functions v ∈ C
k(Ω) that vanish outside
some compact subset of Ω, in particular, such functions satisfy Dαv = 0 on
the boundary of Ω for |α| ≤ k. Similarly, C∞0 (R
d) is the set of functions that
have continuous derivatives of all orders and vanish outside some bounded
set.
We say that a function is smooth if, depending on the situation, it has
sufficiently many continuous derivatives.
We also frequently employ the space L2(Ω) of square integrable functions
with scalar product and norm

1.3 Physical Derivation of the Heat Equation 7
(v, w) = (v, w)L2(Ω) =
∫
Ω
vw dx, ∥v∥ = ∥v∥L2(Ω) =
(∫
Ω
v2 dx
)1/2
.
For Ω a domain we also employ the Sobolev space Hk(Ω), k ≥ 1, of functions
v such that Dαv ∈ L2(Ω) for all |α| ≤ k, equipped with the norm and
seminorm
∥v∥k = ∥v∥Hk(Ω) =
( ∑
|α|≤k
∥Dαv∥2
)1/2
,
|v|k = |v|Hk(Ω) =
( ∑
|α|=k
∥Dαv∥2
)1/2
.
Additional norms are defined and used locally when the need arises.
We use the letters c, C to denote various positive constants that need not
be the same at each occurrence.
1.3 Physical Derivation of the Heat Equation
Many equations in physics are derived by combining a conservation law with
constitutive relations. A conservation law states that a physical quantity,
such as energy, mass, or momentum, is conserved as the physical process
develops in time. Constitutive relations express our assumptions about how
the material behaves when the state variables change.
In this section we we consider the conduction of heat in a body Ω ⊂ R3
with boundary Γ and derive the heat equation using conservation of energy
together with linear constitutive relations.
Conservation of Energy
Consider the balance of heat in an arbitrary subset Ω0 ⊂ Ω with boundary
Γ0. The energy principle says that the rate of change of the total energy in
Ω0 equals the inflow of heat through Γ0 plus the heat power produced by
heat sources inside Ω0. To express this in mathematical terms we introduce
some physical quantities, each of which is followed, within brackets, by the
associated standard unit of measurement.
With e = e(x, t) [J/m3] the density of internal energy at the point x [m]
and time t [s], the total amount of heat in Ω0 is
∫
Ω0
e dx [J]. Further with
the vector field j = j(x, t) [J/(m2s)] denoting the heat flux and n the exterior
unit normal to Γ0, the net outflow of heat through Γ0 is
∫
Γ0
j · n ds [J/s].
Introducing also the power density of heat sources p = p(x, t) [J/(m3s)], the
energy principle then states that
d
dt
∫
Ω0
e dx = −
∫
Γ0
j · n ds +
∫
Ω0
p dx.

8 1 Introduction
Applying the divergence theorem we obtain
∫
Ω0
(
∂e
∂t
+ ∇ · j − p
)
dx = 0, for t > 0.
Since Ω0 ⊂ Ω is arbitrary this implies
(1.10)
∂e
∂t
+ ∇ · j = p in Ω, for t > 0.
Constitutive Relations
The internal energy density e depends on the absolute temperature T [K]
and the spatial coordinates, and in our first constitutive relation we assume
that e depends linearly on T near a suitably chosen reference temperature
T0, that is,
(1.11) e = e0 + σ(T − T0) = e0 + σ ϑ, where ϑ = T − T0.
The coefficient σ = σ(x) [J/(m3 K)] is called the specific heat capacity. (It is
usually expressed in the form σ = ρ c, where ρ [kg/m3] is mass density and
c [J/(kg K)] is the specific heat capacity per unit mass.)
According to Fourier’s law the heat flux due to conduction is proportional
to the temperature gradient, which gives a second constitutive relation,
j = −λ∇ϑ.
The coefficient λ = λ(x) [J/(m K s)] is called the heat conductivity. In some
situations (e.g., gas in a porous medium, heat transport in a fluid) heat is
also transported by convection with heat flux v e, where v = v(x, t) [m/s] is
the convective velocity vector field. The constitutive relation then reads
(1.12) j = −λ∇ϑ + v e.
Substituting (1.11) and (1.12) into (1.10) we obtain
(1.13) σ
∂ϑ
∂t
− ∇ · (λ∇ϑ) + ∇ · (σ v ϑ) = q in Ω, where q = p − ∇ · (v e0),
which is the heat equation with convection.
Boundary Conditions
In the modelling of heat conduction, the differential equation (1.13) is com-
bined with an initial condition at time t = 0,
(1.14) ϑ(x, 0) = ϑi(x),

1.3 Physical Derivation of the Heat Equation 9
and a boundary condition, expressing that the heat flux through the boundary
is proportional to the difference between the surface temperature and the
ambient temperature, j · n = κ(ϑ − ϑa), where κ = κ(x, t) [J/(m
2 s K)] is a
heat transfer coefficient. Assuming that the material flow does not penetrate
the boundary, i.e., v · n = 0, we obtain from (1.12)
j · n = −λ∇ϑ · n = −λ
∂ϑ
∂n
on Γ,
where ∂ϑ/∂n = ∇ϑ · n denotes the exterior normal derivative of ϑ. Therefore
the boundary condition is Robin’s boundary condition
(1.15) λ
∂ϑ
∂n
+ κ(ϑ − ϑa) = 0 on Γ.
The limit case κ = 0 means that the boundary surface is perfectly insulated,
so that we have Neumann’s boundary condition,
∂ϑ
∂n
= 0.
At the other extreme, dividing by κ in (1.15) and letting κ → ∞, we obtain
Dirichlet’s boundary condition
(1.16) ϑ = ϑa.
The limit case κ = ∞ thus means that the body is in perfect thermal contact
with the surroundings, i.e., heat flows freely through the surface, so that the
surface temperature of the body is equal to the ambient temperature.
Dimensionless Form
It is often useful to write the above equations in dimensionless form. Choosing
reference constants L [m], τ [s], ϑf [K], σf [J/(m
3 K)], vf [m/s], etc., we
define dimensionless variables
t̃ = t/τ, x̃ = x/L, u(x̃, t̃) = ϑ(x̃L, t̃τ )/ϑf.
In order to make the heat equation (1.13) dimensionless we divide it by
λfϑf/L
2. Using the chain rule,
∂u
∂t̃
= τ
∂
∂t
( ϑ
ϑf
)
, ∇̃u = L∇
( ϑ
ϑf
)
,
we get
(1.17) d
∂u
∂t̃
− ∇̃ · (a∇̃u) + ∇̃ · (bu) = f in Ω̃,
where

10 1 Introduction
d =
L2σf
τ λf
σ
σf
, a =
λ
λf
, b =
vfσfL
λf
σ
σf
v
vf
, f =
L2
λfϑf
q.
It is natural to choose τ = L2σf/λf, so that d = 1 if σ = σf is constant. The
dimensionless number Pe = vfσfL/λf that appears in the definition of b is
called Peclet’s number and measures the relative strengths of convection and
conduction. Skipping the tilde from now on, we write (1.17) as
(1.18) d
∂u
∂t
− ∇ · (a∇u) + b · ∇u + cu = f in Ω, where c = ∇ · b.
The boundary condition (1.15) and the initial condition (1.14) transform
in a similar way to
(1.19) a
∂u
∂n
+ h(u − ua) = 0 on Γ,
and
(1.20) u(x, 0) = ui(x).
Here h = Bi κ/κf, where Bi = Lκf/λf is called the Biot number.
The partial differential equation (1.18) together with the initial condition
(1.20) and the boundary condition (1.19) is called an initial-boundary value
problem. The term −∇·(a∇u) is written in divergence form. This form arises
naturally in the derivation of the equation, and it is convenient in much of
the mathematical analysis, as we shall see below. However, we sometimes
expand the derivative and write the equation in non-divergence form:
(1.21) d
∂u
∂t
− a∆u + b̄ · ∇u + cu = f, where b̄ = b − ∇a.
Some Simplified Problems
It is useful to study various simplifications of the above equations, because
it may then be possible to carry the mathematical analysis further than in
the general case. If we assume that the coefficients are constant, with b = 0,
c = 0, then (1.18) reduces to (recall that d = 1 if σ is constant)
(1.22)
∂u
∂t
− a∆u = f.
For a = 1 this is equation (1.6). If f and the boundary condition are indepen-
dent of t, then u could be expected to approach a stationary state as t grows,
i.e., u(x, t) → v(x) as t → ∞, and since we should then have ∂u/∂t → 0, we
find that v satisfies Poisson’s equation (1.1). If in addition f = 0, we have
Laplace’s equation
−∆u = 0.

1.3 Physical Derivation of the Heat Equation 11
Solutions of Laplace’s equation are called harmonic functions.
Another important kind of simplification is obtained by reduction of di-
mension. For example, consider stationary (time-independent) heat conduc-
tion in a (not necessarily circular) cylinder oriented along the x1-axis with
insulated mantle surface. If the coefficients a, b, c, f in (1.18) are independent
of x2 and x3, then it is reasonable to assume that the solution u also depends
only on one variable x1, which we then denote by x, i.e., u = u(x). The heat
equation (1.18) then reduces to an ordinary differential equation
−(au′)′ + bu′ + cu = f in Ω = (0, 1).
The boundary condition (1.19) becomes
(1.23) −a(0)u′(0) + h0(u(0) − u0) = 0, a(1)u
′(1) + h1(u(1) − u1) = 0.
We call this a two-point boundary value problem. Similar simplifications are
obtained under cylindrical and spherical symmetry by writing the equations
in cylindrical respectively spherical coordinates. If the coefficients are con-
stant, then we can readily express the solution in terms of well-known special
functions, see Problem 1.6.
Nonlinear Equations, Linearization
The coefficients in the heat equation (1.18) and in the boundary conditions
often depend on the temperature u, which makes the equations nonlinear.
Although the study of nonlinear equations is outside the scope of this book,
we mention that the study of nonlinear equations often proceeds by lineariza-
tion, i.e., by reduction to the study of related linear equations. We illustrate
this in the case of the equation
F (u) :=
∂u
∂t
− ∇ ·
(
a(u)∇u
)
− f (u) = 0 in Ω, for t > 0,
which is of the form (1.18), and which is to be solved together with suitable
initial and boundary conditions. One approach to such a problem is to use
Newton’s method, which produces a sequence of approximate solutions uk
from a starting guess u0 in the following way: Given uk we want to find an
increment vk such that uk+1 = uk + vk is a better approximation of the exact
solution than uk. Approximating F (uk+1) = 0 by F (uk) + F ′(uk)vk = 0, we
obtain a linearized equation
∂vk
∂t
− ∇ ·
(
a(uk)∇vk
)
− ∇ ·
(
a′(uk)∇ukvk
)
− f ′(uk)vk = −F (uk) in Ω,
which is solved together with an initial condition and linearized boundary
conditions. This equation is a linear equation in vk of the form (1.18), where
the new coefficients a(uk(x, t)), etc., depend on x and t.

12 1 Introduction
1.4 Problems
Problem 1.1. (Derivation of the convection-diffusion equation.) Let c =
c(x, t) [mol/m3] denote the concentration at the point x [m] and time t
[s] of a substance that is being transported through a domain x ∈ Ω ⊂ R3
by convection and diffusion. The flux due to convection is
jc = vc, [mol/(m
2s)]
where v = v(x) [m/s] is the convective velocity field. The flux due to diffusion
is (Fick’s law)
jd = −D∇c, [mol/(m
2s)]
where D = D(x) [m2/s] is the diffusion coefficient. Let r [mol/(m3s)] denote
the rate of creation/annihilation of material, e.g., by chemical reaction. The
total mass of the substance within an arbitrary subdomain is
∫
Ω0
c dx. Use
the conservation of mass and the divergence theorem to derive the convection-
diffusion equation
∂c
∂t
− ∇ · (D∇c) + ∇ · (vc) = r, [mol/(m3s)]
which is of the same mathematical form as (1.13). Derive a boundary con-
dition of the form (1.15). Show that these equations can be written in the
same dimensionless form as (1.18) and (1.19).
Problem 1.2. (Derivation of the wave equation.) Consider the longitudinal
motion of an elastic bar of length L [m]and of constant cross-sectional area A
[m2] and with density ρ [kg/m3]. Let u = u(x, t) [m]denote the displacement
at time t [s] of a cross-section originally located at x ∈ [0, L]. Newton’s law
of motion states that
d
dt
∫ b
a
pA dx =
(
σ(b) − σ(a)
)
A, [N]
where
∫ b
a
pA dx [kg m/s] is the total momentum of an arbitrary segment
(a, b) and σ [N/m2] is the stress (force per unit cross-sectional area). This
leads to
∂p
∂t
=
∂σ
∂x
.
For small displacements we have a linear relationship between the stress σ
and the strain ϵ = ∂u/∂x, namely Hooke’s law,
σ = Eϵ,
where E [N/m2] is the modulus of elasticity, and the momentum density is
given by p = ρ∂u/∂t. Show that u satisfies the wave equation

1.4 Problems 13
ρ
∂2u
∂t2
=
∂
∂x
(
E
∂u
∂x
)
.
Discuss various possible boundary conditions at the ends of the bar. For
example, at x = L:
– fixed end, u(L) = 0,
– free end, σ(L) = 0, which leads to ux(L) = 0,
– elastic support, σ(L) = −ku(L), which leads to Eux(L) + ku(L) = 0.
Note that these are of the form (1.23).
Problem 1.3. (Elastic beam.) Consider the bending of an elastic beam that
extends along the interval 0 ≤ x ≤ L. At an arbitrary cross-section at a
distance x from the left end we introduce the bending moment (torque) M =
M (x) [Nm], the transversal force T = T (x) [N], and the external applied
force q = q(x) per unit length [N/m]. It can be shown that equilibrium
of forces requires M ′ = T and T ′ = −q. Let u = u(x) [m] be the small
transversal deflection of the beam. The bending angle is then approximately
u′. The constitutive law is M = −EIu′′, where E [N/m2] the modulus of
elasticity and I [m4] is a moment of inertia of the cross-section of the beam.
Show that this leads to the fourth order equation
(EIu′′)′′ = q.
Discuss various possible boundary conditions at the ends of the beam. For
example, at x = L:
– clamped end, u(L) = 0, u′(L) = 0,
– free end, M (L) = −(EIu′′)(L) = 0, T (L) = −(EIu′′)′(L) = 0,
– hinge, u′(L) = 0, M (L) = −(EIu′′)(L) = 0.
Problem 1.4. (The Laplace operator in spherical symmetry.) Introduce
spherical coordinates (r, θ, φ) defined by x1 = r sin θ cos φ, x2 = r sin θ sin φ,
x3 = r cos θ. Assume that the function u does not depend on θ and φ, i.e.,
u = u(r). Show that
∆u =
1
r2
d
dr
(
r2
du
dr
)
.
Problem 1.5. (The Laplace operator in cylindrical symmetry.) Introduce
cylindrical coordinates (ρ, ϕ, z) defined by x1 = ρ cos ϕ, x2 = ρ sin ϕ, x3 = z.
Assume that the function u does not depend on ϕ and z, i.e., u = u(ρ). Show
that
∆u =
1
ρ
d
dρ
(
ρ
du
dρ
)
.
Problem 1.6. Let Ω = {x ∈ R3 : |x| < 1}. Determine an explicit solution of the boundary value problem −∆u + c2u = f in Ω, with u = g on Γ, 14 1 Introduction assuming spherical symmetry and that c, f, g are constants. That is, solve −(r2u′(r))′ + c2r2u(r) = r2f for r ∈ (0, 1), with u(1) = g, u(0) finite. Hint: Set v(r) = ru(r). 2 A Two-Point Boundary Value Problem For the purpose of preparing for the treatment of boundary value problems for elliptic partial differential equations we consider here a simple two-point boundary value problem for a second order linear ordinary differential equa- tion. In the first section we derive a maximum principle for this problem, and use it to show uniqueness and continuous dependence on data. In the second section we construct a Green’s function in a special case and show how this implies the existence of a solution. In the third section we write the problem in variational form, and use this together with simple tools from functional analysis to prove existence, uniqueness, and continuous dependence on data. 2.1 The Maximum Principle We consider the boundary value problem (2.1) Au := − (au′)′ + bu′ + cu = f in Ω = (0, 1), u(0) = u0, u(1) = u1, where the coefficients a = a(x), b = b(x), and c = c(x) are smooth functions with (2.2) a(x) ≥ a0 > 0, c(x) ≥ 0, for x ∈ Ω̄ = [0, 1],
and where the function f = f (x) and the numbers u0, u1 are given, cf.
Sect. 1.3.
In the particular case that a = 1, b = c = 0, this reduces to
(2.3) −u′′ = f in Ω, with u(0) = u0, u(1) = u1.
By integrating this differential equation twice we find that a solution must
have the form
(2.4) u(x) = −
∫ x
0
∫ y
0
f (s) ds dy + αx + β,
with the constants α, β to be determined. Setting x = 0 and x = 1 we find

16 2 A Two-Point Boundary Value Problem
α = u1 − u0 +
∫ 1
0
∫ y
0
f (s) ds dy, β = u0.
Reversing the steps we find that (2.4), with these α, β, is the unique solution
of (2.3).
In the special case f = 0 the solution of (2.3) is the linear function
u(x) = u0(1 − x) + u1x. In particular, the values of this function lie between
those at x = 0 and x = 1, and its maximum and minimum are thus located
at the endpoints of the interval Ω. More generally, we have the following
maximum (minimum) principle for (2.1).
Theorem 2.1. Consider the differential operator A in (2.1), and assume
that u ∈ C2 = C2(Ω̄) and
(2.5) Au ≤ 0
(
Au ≥ 0
)
in Ω.
(i) If c = 0, then
(2.6) max
Ω̄
u = max
{
u(0), u(1)
} (
min
Ω̄
u = min
{
u(0), u(1)
})
.
(ii) If c ≥ 0 in Ω, then
(2.7) max
Ω̄
u ≤ max
{
u(0), u(1), 0
} (
min
Ω̄
u ≥ min
{
u(0), u(1), 0
})
.
In case (i) we conclude that the maximum of u is attained at the bound-
ary, i.e., at one of the endpoints of the interval Ω. In case (ii) we draw the
same conclusion if the maximum is nonnegative. This does not exclude the
possibility that the maximum is attained also in the interior of Ω. However,
there is also a stronger form of the maximum principle, which in case (i)
asserts that if (2.5) holds and u has a maximum at an interior point of Ω (in
case (ii) a nonnegative interior maximum), then u is constant in Ω̄. We shall
not prove this here, but we refer to Sect. 3.3 below for the corresponding
result for harmonic functions. The variants within parentheses, with Au ≥ 0,
may be described as a minimum principle; it is reduced to the maximum
principle by looking at −u.
Proof. (i) Assume first, instead of (2.5), that Au < 0 in Ω. If u has a maxi- mum at an interior point x0 ∈ Ω, then at this point we have u ′(x0) = 0 and u′′(x0) ≤ 0, so that Au(x0) ≥ 0, which contradicts our assumption. Hence u cannot have an interior maximum point and (2.6) follows. Assume now that we only know that Au ≤ 0 in Ω. Let φ be a function such that φ ≥ 0 in Ω̄ and Aφ < 0 in Ω. For example, we may use the function φ(x) = eλx with λ so large that Aφ = (−aλ2 + (b − a′)λ)φ < 0 in Ω̄. Assume now that u attains its maximum at an interior point x0 but not at x = 0 or x = 1. Then for ϵ > 0 sufficiently small this is true also for v = u + ϵφ. But
Av = Au + ϵAφ < 0 in Ω̄, which contradicts the first part of the proof. 2.1 The Maximum Principle 17 (ii) If u ≤ 0 in Ω, then (2.7) holds trivially. Otherwise assume that maxΩ̄ u = u(x0) > 0 and x0 ̸= 0, 1. Let (α, β) be the largest subinterval
of Ω containing x0 in which u > 0. We now have Ãu := Au − cu ≤ 0 in
(α, β). Part (i), applied with the operator à in the interval (α, β), therefore
implies u(x0) = max{u(α), u(β)}. But then α and β could not both be in-
terior points of Ω, for then either u(α) or u(β) would be positive, and the
interval (α, β) would not be as large as possible with u > 0. This implies
u(x0) = max{u(0), u(1)} and hence (2.7). ⊓.
As a consequence of this theorem we have the following stability estimate
with respect to the maximum-norm, where we use the notation of Sect. 1.2.
Theorem 2.2. Let A be as in (2.1) and (2.2). If u ∈ C2, then
∥u∥C ≤ max
{
|u(0)|, |u(1)|
}
+ C∥Au∥C.
The constant C depends on the coefficients of A but not on u.
Proof. We shall bound the maxima of ±u. We set φ(x) = eλ − eλx and define
the two functions
v±(x) = ±u(x) − ∥Au∥C φ(x).
Since φ ≥ 0 in Ω and Aφ = ceλ + (aλ2 + (a′ − b)λ − c)eλx ≥ 1 in Ω̄, if λ > 0
is chosen sufficiently large, we have, with such a choice of λ,
Av± = ±Au − ∥Au∥CAφ ≤ ±Au − ∥Au∥C ≤ 0 in Ω.
Theorem 2.1(ii) therefore yields
max
Ω̄
(v±) ≤ max
{
v±(0), v±(1), 0
}
≤ max
{
± u(0), ±u(1), 0
}
≤ max
{
|u(0)|, |u(1)|
}
,
because v±(x) ≤ ±u(x) for all x. Hence,
max
Ω̄
(±u) = max
Ω̄
(
v± + ∥Au∥C φ
)
≤ max
Ω̄
(v±) + ∥Au∥C∥φ∥C
≤ max
{
|u(0)|, |u(1)|
}
+ C∥Au∥C, with C = ∥φ∥C,
which completes the proof. ⊓.
From Theorem 2.2 we immediately conclude the uniqueness of a solution
of (2.1). In fact, if u and v were two solutions, then their difference w = u−v
would satisfy Aw = 0, w(0) = w(1) = 0, and hence ∥w∥C = 0, so that u = v.
More generally, if u and v are two solutions of (2.1) with right hand sides
f and g and boundary values u0, u1 and v0, v1, respectively, then
∥u − v∥C ≤ max
{
|u0 − v0|, |u1 − v1|
}
+ C∥f − g∥C.

18 2 A Two-Point Boundary Value Problem
Thus the problem (2.1) is stable, i.e., a small change in data does not cause
a big change in the solution.
As another application of the maximum principle we note that if all the
data of the boundary value problem (2.1) are nonpositive, then the solution
is nonpositive. That is, if f ≤ 0 and u0, u1 ≤ 0, then u ≤ 0. By means of the
stronger variant of the maximum principle mentioned after Theorem 2.1, we
may even conclude that u < 0 in Ω unless u(x) ≡ 0. More generally, we have the following monotonicity property: If Au = f in Ω, with u(0) = u0, u(1) = u1, Av = g in Ω, with v(0) = v0, v(1) = v1, and if f ≤ g, u0 ≤ v0, and u1 ≤ v1, then u ≤ v. 2.2 Green’s Function We now consider the problem (2.1) with b = 0 and with boundary values u0 = u1 = 0. We shall derive a representation of a solution in terms of a so-called Green’s function G(x, y). For this purpose, let U0 and U1 be two solutions of the homogeneous equation such that AU0 = 0 in Ω, with U0(0) = 1, U0(1) = 0, AU1 = 0 in Ω, with U1(0) = 0, U1(1) = 1. To see that such solutions exist, we note that by the standard theory of ordinary differential equations the initial value problem for Au = 0 with u(0) = 0, u′(0) = 1 has a unique solution, and that u(1) ̸= 0 for this solution, since otherwise u(x) ≡ 0 in Ω by Theorem 2.2. By multiplication of this solution by an appropriate constant we obtain the desired function U1. The function U0 is constructed similarly, starting at x = 1. By Theorem 2.1 U0 and U1 are nonnegative. We refer to Problem 2.5 for the case when b ̸= 0. Theorem 2.3. Let b = 0 and let U0, U1 be as described above. Then a solution of (2.1) with u0 = u1 = 0 is given by (2.8) u(x) = ∫ 1 0 G(x, y)f (y) dy, where G(x, y) = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 1 κ U0(x)U1(y), for 0 ≤ y ≤ x ≤ 1, 1 κ U1(x)U0(y), for 0 ≤ x ≤ y ≤ 1, and (2.9) κ = a(x) ( U0(x)U ′ 1(x) − U ′ 0(x)U1(x) ) ≡ constant > 0.

2.2 Green’s Function 19
Proof. We begin by showing that κ is constant: Since (a U ′j )
′ = c Uj , we have
κ′ = U0(a U
′
1)
′
− U1(a U
′
0)
′ = U0 c U1 − U1 c U0 = 0.
Setting x = 0 we find κ = a(0) U ′1(0) ̸= 0, because otherwise U1(0) = U
′
1(0) =
0 and hence U1(x) ≡ 0. Since U1 is nonnegative we have U
′
1(0) nonnegative
and hence it follows that κ > 0.
Clearly u as defined in (2.8) satisfies the homogeneous boundary condi-
tions. To show that it is a solution of the differential equation we write
u(x) =
∫ x
0
G(x, y)f (y) dy +
∫ 1
x
G(x, y)f (y) dy
=
1
κ
U0(x)
∫ x
0
U1(y)f (y) dy +
1
κ
U1(x)
∫ 1
x
U0(y)f (y) dy.
Hence, by differentiation,
u′(x) =
1
κ
(
U ′0(x)
∫ x
0
U1(y)f (y) dy + U0(x)U1(x)f (x)
)
+
1
κ
(
U ′1(x)
∫ 1
x
U0(y)f (y) dy − U1(x)U0(x)f (x)
)
,
where the terms involving f (x) cancel. Multiplying by −a(x) and differenti-
ating we thus obtain, using (a U ′j )
′ = c Uj and (2.9),
−(a(x)u′(x))′ = −
1
κ
(a(x)U ′0(x))
′
∫ x
0
U1(y)f (y) dy
−
1
κ
(a(x)U ′1(x))
′
∫ 1
x
U0(y)f (y) dy
−
1
κ
a(x)
(
U ′0(x)U1(x) − U
′
1(x)U0(x)
)
f (x)
= −
1
κ
c(x)U0(x)
∫ x
0
U1(y)f (y) dy
−
1
κ
c(x)U1(x)
∫ 1
x
U0(y)f (y) dy + f (x)
= −c(x)
∫ 1
0
G(x, y)f (y) dy + f (x) = −c(x)u(x) + f (x),
which completes the proof. ⊓.
In particular, this theorem shows the existence of a solution of the problem
considered. We already know from Sect. 2.1 that the solution is unique. The
representation of the solution as an integral in terms of the Green’s function
can also be used to obtain additional information about the solution. As a
simple example we have the maximum-norm estimate

20 2 A Two-Point Boundary Value Problem
(2.10) ∥u∥C ≤ C∥f∥C, with C = max
x∈Ω̄
∫ 1
0
G(x, y) dy,
which gives a more precise value of the constant in Theorem 2.2. Here we have
used the fact that U0 and U1, and hence G, are nonnegative by Theorem 2.1.
Theorem 2.3 may also be used to show the existence of a solution for
general boundary values u0 and u1. In fact, if ū(x) = u0(1 − x) + u1x, and if
v is a solution of
Av = g := f − Aū in Ω, with v(0) = v(1) = 0,
then u = v + ū satisfies Au = f and u(0) = u0, u(1) = u1.
2.3 Variational Formulation
We shall now treat our two-point boundary value problem within the frame-
work of the Hilbert space L2 = L2(Ω), and derive a so-called variational
formulation. We refer to App. A for the functional analytic concepts used.
We consider the boundary value problem (2.1) with homogeneous bound-
ary conditions, i.e.,
(2.11) Au := −(au′)′ + bu′ + cu = f in Ω = (0, 1), with u(0) = u(1) = 0.
We assume that the coefficients a, b, and c are smooth and, instead of (2.2),
that
(2.12) a(x) ≥ a0 > 0, c(x) − b
′(x)/2 ≥ 0, for x ∈ Ω̄.
Multiplying the differential equation by a function ϕ ∈ C10 = C
1
0 (Ω), and
integrating over the interval Ω, we obtain
(2.13)
∫ 1
0
(
− (au′)′ + bu′ + cu
)
ϕ dx =
∫ 1
0
f ϕ dx,
or, after integration by parts, using ϕ(0) = ϕ(1) = 0,
(2.14)
∫ 1
0
(au′ϕ′ + bu′ϕ + cuϕ) dx =
∫ 1
0
f ϕ dx, ∀ϕ ∈ C10 ,
which we refer to as the variational or weak formulation of (2.11).
Introducing the bilinear form
(2.15) a(v, w) =
∫ 1
0
(av′w′ + bv′w + cvw) dx,
and the linear functional

2.3 Variational Formulation 21
L(w) = (f, w) =
∫ 1
0
f w dx,
and using the fact that C10 is dense in H
1
0 = H
1
0 (Ω), we may write the equation
(2.14) as
(2.16) a(u, ϕ) = L(ϕ), ∀ϕ ∈ H10 .
We say that u is a weak solution of (2.11) if u ∈ H10 and (2.16) holds.
Thus we do not require a weak solution to be twice differentiable. However, if
a weak solution belongs to C2, then it is actually a classical solution of (2.11).
In fact, by integration by parts in (2.14) we conclude that (2.13) holds, i.e.,
∫ 1
0
(Au − f ) ϕ dx = 0, ∀ϕ ∈ H10 .
This immediately implies Au = f in Ω, and since u ∈ H10 we also have
u(0) = u(1) = 0. This calculation can also be performed if u ∈ H2 ∩ H10 , in
which case we say that u is a strong solution of (2.11).
We note that, with the notation of Sect. 1.2,
(2.17) ∥v∥ ≤ ∥v′∥, if v(0) = v(1) = 0.
In fact, by the Cauchy-Schwarz inequality we have for all x ∈ Ω,
|v(x)|2 =
∣
∣
∣
∫ x
0
v′(y) dy
∣
∣
∣
2
≤
∫ x
0
12 dy
∫ x
0
(v′)2 dy ≤ x
∫ 1
0
(v′)2 dy ≤ ∥v′∥2,
from which (2.17) follows by integration. This is a special case of Poincaré’s
inequality, which has a counterpart also for functions of several variables, see
Theorem A.6. It follows at once that
(2.18) ∥v∥1 =
(
∥v∥2 + ∥v′∥2
)1/2
≤
√
2∥v′∥, ∀v ∈ H10 ,
which shows that ∥v∥1 and |v|1 = ∥v
′∥ are equivalent norms.
Using our assumption (2.12), we find that
∫ 1
0
(bv′v + cv2) dx =
[
1
2
b v2
]1
0
+
∫ 1
0
(c − 1
2
b′) v2 dx ≥ 0, for v ∈ H10 .
Hence, from (2.12) and (2.18) it follows that the bilinear form a(v, w) has the
property
(2.19) a(v, v) ≥ min
x∈Ω̄
a(x) ∥v′∥2 ≥ α∥v∥21, ∀v ∈ H
1
0 , with α = a0/2 > 0.
The inequality (2.19) expresses that the bilinear form a(·, ·) is coercive in H10 ,
see (A.12). Setting ϕ = u in (2.16) and using (2.19) and (2.17), we find

22 2 A Two-Point Boundary Value Problem
α∥u∥21 ≤ a(u, u) = (f, u) ≤ ∥f∥ ∥u∥ ≤ ∥f∥ ∥u∥1,
so that
(2.20) ∥u∥1 ≤ C∥f∥, with C = 2/a0.
The bilinear form a(v, w) is also bounded on H10 in the sense that (cf. (A.9))
(2.21) |a(v, w)| ≤ C∥v∥1∥w∥1, ∀v, w ∈ H
1
0 .
For, estimating the coefficients in (2.15) by their maxima and using the
Cauchy-Schwarz inequality, we have
|a(v, w)| ≤ C
∫ 1
0
(
|v′w′| + |v′w| + |vw|
)
dx ≤ C∥v∥1∥w∥1.
We now turn to the question of existence of a solution of the variational
equation (2.16).
Theorem 2.4. Assume that (2.12) holds and let f ∈ L2. Then there exists
a unique solution u ∈ H10 of (2.16). This solution satisfies (2.20).
Proof. The proof is based on the Lax-Milgram lemma, Theorem A.3. We al-
ready checked that a(·, ·) is coercive and bounded in H10 . The linear functional
L(·) is also bounded in H10 , because
|L(ϕ)| = |(f, ϕ)| ≤ ∥f∥ ∥ϕ∥ ≤ ∥f∥ ∥ϕ∥1, ∀ϕ ∈ H
1
0 .
Hence the assumptions of the Lax-Milgram lemma are satisfied and it follows
that there exists a unique u ∈ H10 satisfying (2.16). Together with (2.20) this
completes the proof. ⊓.
We remark that when b = 0 the bilinear form a(·, ·) is symmetric positive
definite and thus an inner product, with the associated norm equivalent to
∥·∥1. The existence of a unique solution then follows from the more elementary
Riesz representation theorem, Theorem A.1.
In the symmetric case when b = 0, the solution of (2.16) may also be
characterized as the minimizer of a certain quadratic functional, see Theorem
A.2. This is a special case of the famous Dirichlet principle.
Theorem 2.5. Assume that (2.2) holds and that b = 0. Let f ∈ L2 and
u ∈ H10 be the solution of (2.16), and set
F (ϕ) = 1
2
∫ 1
0
(
a(ϕ′)2 + cϕ2
)
dx −
∫ 1
0
f ϕ dx.
Then F (u) ≤ F (ϕ) for all ϕ ∈ H10 , with equality only for ϕ = u.

2.4 Problems 23
The weak solution u of (2.16) obtained in Theorem 2.4 is actually more
regular than stated there. Using our definitions one may, in fact, show that u′′
exists as a weak derivative (cf. (A.21)), and that au′′ = −f + (b−a′)u′ + cu ∈
L2. It follows that u ∈ H
2 and that
a0∥u
′′
∥ ≤ ∥au′′∥ ≤ ∥f∥ + ∥(b − a′)u′∥ + ∥cu∥ ≤ ∥f∥ + C∥u∥1 ≤ C∥f∥.
Together with (2.20) this implies the regularity estimate
(2.22) ∥u∥2 ≤ C∥f∥.
We conclude that the weak solution of (2.1) found in Theorem 2.4 is actually
a strong solution. The proof of H2-regularity uses the assumption that a is
smooth and f ∈ L2. With a less smooth, or with f only in H
−1, see (A.30),
we still obtain a weak solution in H10 , but then it may not belong to H
2, see
Problem 2.8.
2.4 Problems
Problem 2.1. Determine explicit solutions of the boundary value problem
−u′′ + cu = f in (−1, 1), with u(−1) = u(1) = g,
where c, f, g are constants. Use this to illustrate the maximum principle.
Problem 2.2. Determine Green’s functions for the following problems:
− u′′ = f in Ω = (0, 1), with u(0) = u(1) = 0,(a)
− u′′ + cu = f in Ω = (0, 1), with u(0) = u(1) = 0,(b)
where c is a positive constant.
Problem 2.3. Consider the nonlinear boundary value problem
−u′′ + u = eu in Ω = (0, 1), with u(0) = u(1) = 0.
Use the maximum principle to show that all solutions are nonnegative, i.e.,
u(x) ≥ 0 for all x ∈ Ω̄. Use the strong version of the maximum principle to
show that all solutions are positive, i.e., u(x) > 0 for all x ∈ Ω.
Problem 2.4. Assume that b = 0 as in Theorem 2.3 and let G(x, y) be the
Green’s function defined there.
(a) Prove that G is symmetric, G(x, y) = G(y, x).
(b) Prove that
a(v, G(x, ·)) = v(x), ∀v ∈ H10 , x ∈ Ω.
This means that AG(x, ·) = δx, where δx is Dirac’s delta at x, defined as
the linear functional δx(φ) = φ(x) for all φ ∈ C
0
0 , see Problem A.9.

24 2 A Two-Point Boundary Value Problem
Problem 2.5. In the unsymmetric case when b ̸= 0, Green’s function is
defined in a similar way as in Theorem 2.3:
G(x, y) =
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
U0(x)U1(y)
κ(y)
, for 0 ≤ y ≤ x ≤ 1,
U1(x)U0(y)
κ(y)
, for 0 ≤ x ≤ y ≤ 1.
The main difference is that κ is no longer constant. The functions U0 and
U1 are linearly independent, and hence it follows from the theory of ordinary
differential equations that their Wronski determinant U0U
′
1 − U
′
0U1 does not
vanish. As before we may then conclude that κ(x) > 0 in Ω̄. Repeat the steps
of the proof Theorem 2.3 in this case.
Problem 2.6. Give variational formulations and prove existence of solutions
of
−u′′ = f in Ω = (0, 1),
with the following boundary conditions
(a) u(0) = u(1) = 0,
(b) u(0) = u′(1) = 0,
(c) −u′(0) + u(0) = u′(1) = 0.
Problem 2.7. Consider the “beam equation” from Problem 1.3,
d4u
dx4
= f in Ω = (0, 1),
together with the boundary conditions
(a) u(0) = u′(0) = u(1) = u′(1) = 0,
(b) u(0) = u′′(0) = u(1) = u′′(1) = 0,
(c) u(0) = u′(0) = u′(1) = u′′′(1) = 0,
(d) u(0) = u′(0) = u′′(1) = u′′′(1) = 0,
(e) u(0) = u′(0) = u(1) = u′′′(1) = 0.
Give variational formulations and investigate existence and uniqueness of
solutions of these problems. Give mechanical interpretations of the boundary
conditions.
Problem 2.8. Find an explicit solution of (2.11) with a = 1, b = c = 0, and
f (x) = 1/x. Recall from Problem A.11 that f ∈ H−1 but f ̸∈ L2. Check that
u ∈ H10 but u ̸∈ H
2. Hint: u(x) = −x log x.

3 Elliptic Equations
In this chapter we study boundary value problems for elliptic partial differen-
tial equations. As we have seen in Chapt. 1 such equations are central in both
theory and application of partial differential equations; they describe a large
number of physical phenomena, particularly modelling stationary situations,
and are stationary limits of evolution equations. After some preliminaries in
Sect. 3.1 we begin by showing a maximum principle in Sect. 3.2. In the same
way as for the two-point boundary value problem in Chapt. 2 this may be used
to show uniqueness and continuous dependence on data for boundary value
problems. In the following Sect. 3.3 we show the existence of a solution of
Dirichlet’s problem for Poisson’s equation in a disc with homogeneous bound-
ary conditions, using an integral representation in terms of Poisson’s kernel.
In Sect. 3.4 similar ideas are employed to introduce fundamental solutions
of elliptic equations, and we illustrate their use by constructing a Green’s
function. Another important approach, presented in Sect. 3.5, is based on a
variational formulation of the boundary value problem and simple functional
analytic tools. In Sect. 3.6 we discuss briefly the Neumann problem, and in
Sect. 3.7 we describe some regularity results.
3.1 Preliminaries
Rather than considering a general second order elliptic equation of the form
(1.5) we shall restrict ourselves, for the sake of simplicity, to the special case
when the matrix A = (aij ) in (1.5) reduces to a scalar multiple a I of the
identity matrix, where a is a smooth function.
We consider first the Dirichlet problem
(3.1) Au := −∇ ·
(
a∇u
)
+ b · ∇u + cu = f in Ω, with u = g on Γ,
where Ω ⊂ Rd is a domain with appropriately smooth boundary Γ , where
the coefficients a = a(x), b = b(x), c = c(x) are smooth and such that
(3.2) a(x) ≥ a0 > 0, c(x) ≥ 0, ∀x ∈ Ω,
and where f and g are given functions on Ω and Γ , respectively. This is the
stationary case of the heat equation (1.18).

26 3 Elliptic Equations
The particular case a = 1, b = 0, c = 0 is Poisson’s equation, i.e.,
(3.3) −∆u := −
d∑
j=1
∂2u
∂x2j
= f.
When f = 0 this equation is referred to as Laplace’s equation and its solutions
are called harmonic functions.
We note that if v and w are solutions of the two problems
Av = 0 in Ω, with v = g on Γ,
Aw = f in Ω, with w = 0 on Γ,
then u = v + w is a solution of (3.1). It is therefore sometimes convenient to
consider separately the homogeneous equation with given boundary values
and the inhomogeneous equation with vanishing boundary values.
One may also study the partial differential equation in (3.1) together with
Robin’s boundary condition
(3.4) a
∂u
∂n
+ h
(
u − g
)
= 0 on Γ,
where the coefficient h = h(x) is positive and n is the outward unit normal to
Γ . The Dirichlet boundary condition used in (3.1) may be formally obtained
as the extreme case h = ∞ of (3.4). At the other extreme, h = 0, we obtain
Neumann’s boundary condition
∂u
∂n
= 0 on Γ.
Sometimes one considers mixed boundary conditions in which, e.g., Dirichlet
boundary conditions are given on one part of the boundary and Neumann
conditions on the remaining part. A function u ∈ C2(Ω̄) that satisfies the
differential equation and the boundary condition in (3.1) is called a classical
solution of this boundary value problem.
3.2 A Maximum Principle
We begin our study of the Dirichlet problem (3.1) by showing a maximum
principle analogous to that of Theorem 2.1.
Theorem 3.1. Consider the differential operator A in (3.1), and assume
that u ∈ C2 = C2(Ω̄) and
(3.5) Au ≤ 0
(
Au ≥ 0
)
in Ω.
(i) If c = 0, then

3.2 A Maximum Principle 27
(3.6) max
Ω̄
u = max
Γ
u
(
min
Ω̄
u = min
Γ
u
)
.
(ii) If c ≥ 0 in Ω, then
(3.7) max
Ω̄
u ≤ max
{
max
Γ
u, 0
} (
min
Ω̄
u ≥ min
{
min
Γ
u, 0
})
.
Proof. (i) Let φ be a function such that φ ≥ 0 in Ω̄ and Aφ < 0 in Ω. Such a function is, e.g., φ(x) = eλx1 for λ so large that Aφ = (−a λ2 + (b1 − ∂a/∂x1)λ)e λx1 < 0 in Ω̄. Assume now that u attains its maximum at an interior point x0 in Ω but not on Γ . Then for ϵ sufficiently small this is true also for v = u + ϵφ. But Av = Au + ϵAφ < 0 in Ω̄. On the other hand, if the maximum of v is v(x̄0), then ∇v(x̄0) = 0 and hence Av(x̄0) = −a(x̄0)∆v(x̄0) ≥ 0, which is a contradiction, and thus shows our claim. (ii) If u ≤ 0 in Ω, then (3.7) holds trivially. Otherwise assume that maxΩ̄ u = u(x0) > 0 and x0 ∈ Ω. Let Ω0 be the largest open connected
subset of Ω containing x0 in which u > 0. We now have Ãu := Au − cu ≤ 0
in Ω0. Part (i), applied with the operator à in Ω0, therefore implies u(x0) =
maxΓ0 u, where Γ0 is the boundary of Ω0. But then Γ0 could not lie com-
pletely in the open set Ω, for then there would be a point on Γ0 where u were
positive, and Ω0 would not be as large as possible with u > 0. This shows
(3.7). ⊓.
Theorem 3.1 implies stability with respect to the maximum-norm.
Theorem 3.2. Let u ∈ C2(Ω̄). Then there is a constant C such that
∥u∥C(Ω̄) ≤ ∥u∥C(Γ ) + C∥Au∥C(Ω̄).
Proof. Let φ be a function such that φ ≥ 0 and Aφ ≤ −1 in Ω, e.g., a suitable
multiple of the function φ in the proof of Theorem 3.1. We now define two
functions v±(x) = ±u(x) + ∥Au∥C(Ω̄)φ(x). Then
Av± = ±Au + ∥Au∥C(Ω̄)Aφ ≤ 0, in Ω.
Therefore both functions v± take their maxima on Γ , so that
v±(x) ≤ max
Γ
(v±) ≤ max
Γ
(±u) + ∥Au∥C(Ω̄)∥φ∥C(Γ )
≤ ∥u∥C(Γ ) + C∥Au∥C(Ω̄), with C = ∥φ∥C(Γ ).
Since ±u(x) ≤ v±(x) this proves the theorem. ⊓.
In the same way as for the two-point boundary value problem it follows
that there is at most one solution of our Dirichlet problem (3.1), and that, if
uj , j = 1, 2, are solutions of (3.1) with f = fj , g = gj , j = 1, 2, then
∥u1 − u2∥C(Ω̄) ≤ ∥g1 − g2∥C(Γ ) + C∥f1 − f2∥C(Ω̄).

28 3 Elliptic Equations
3.3 Dirichlet’s Problem for a Disc. Poisson’s Integral
In this section we study the Dirichlet problem to find a harmonic function in
a disc Ω = {x ∈ R2 : |x| < R} with given boundary values, i.e., (3.8) − ∆u = 0, for |x| < R, u(R cos ϕ, R sin ϕ) = g(ϕ), for 0 ≤ ϕ < 2π. In the following theorem a solution of (3.8) is given as an integral over the boundary of the disc. Theorem 3.3. (Poisson’s integral formula.) Let PR(r, ϕ) denote the Poisson kernel PR(r, ϕ) = R2 − r2 R2 + r2 − 2rR cos ϕ . Then, using polar coordinates x = (r cos ϕ, r sin ϕ), the function defined by (3.9) u(x) = 1 2π ∫ 2π 0 PR(r, ϕ − ψ)g(ψ) dψ, is a solution of (3.8) for g appropriately smooth, Proof. We first note that, for each n ≥ 0, v(x) = rne±inϕ is a harmonic function. In fact, we have ∆v = ∂2v ∂r2 + 1 r ∂v ∂r + 1 r2 ∂2v ∂ϕ2 , = ( n(n − 1)rn−2 + 1 r nrn−1 − 1 r2 n2rn ) e±inϕ = 0. It follows, for cn bounded, say, that the series (3.10) u(x) = ∞∑ n=−∞ cn ( r R )|n| einϕ is harmonic in Ω. We assume now that g(ϕ) has a Fourier series g(ϕ) = ∞∑ n=−∞ cne inϕ. which is absolutely convergent. The function u(x) in (3.10) with the coef- ficients cn is a then solution of (3.8), and u is continuous in Ω̄. The latter means that u(reiψ) → g(eiϕ) when r → R, ψ → ϕ. To see that this holds, we choose N so large that ∑ |n|>N |cn| < ϵ/3 and write |u(reiψ) − g(eiϕ)| ≤ ∑ |n|≤N |cn| ∣ ∣ ∣ ( r R )|n| einψ − einϕ ∣ ∣ ∣ + 2 ∑ |n|>N
|cn|.

3.3 Dirichlet’s Problem for a Disc. Poisson’s Integral 29
Here obviously the first term on the right tends to 0 when r → R, ψ → ϕ,
and hence becomes smaller than ϵ/3, which shows our claim.
Recall that the Fourier coefficients of g are given by
cn =
1
2π
∫ 2π
0
e−inψg(ψ) dψ.
Formally we thus have
u(x) =
1
2π
∫ 2π
0
∞∑
n=−∞
( r
R
)|n|
ein(ϕ−ψ)g(ψ) dψ,
which is of the form (3.9) with
PR(r, ϕ) =
∞∑
n=−∞
( r
R
)|n|
einϕ.
Setting z = (r/R)eiϕ we have
PR(r, ϕ) = 1 + 2 Re
∞∑
n=1
( r
R
)n
einϕ
= 2 Re
∞∑
n=0
zn − 1 = Re
2
1 − z
− 1 = Re
1 + z
1 − z
= Re
R + reiϕ
R − reiϕ
=
R2 − r2
R2 + r2 − 2rR cos ϕ
,
which completes the proof. ⊓.
One consequence of the theorem is that if u is a harmonic function in Ω,
x̃ is any point in Ω, and if the disc {x : |x − x̃| ≤ R} is contained in Ω, then
(3.11) u(x̃) =
1
2π
∫ 2π
0
u(x̃1 + R cos ψ, x̃2 + R sin ψ) dψ,
since PR(0, ϕ) = 1. Hence u(x̃) is the average of the values of u(x) with
|x − x̃| = R. Thus the value of u at the center of a disc equals the average
of its boundary values. We say that u satisfies the meanvalue property. This
proves a special case of the strong maximum principle we have mentioned
earlier: If a harmonic function u takes its maximum value at an interior point
of Ω, then it is constant. In fact, if x̃ is an interior point of Ω where u attains
its maximum, then by (3.11) u(x) = u(x̃) for all x with {x : |x−x̃| = R} ⊂ Ω,
and since R is arbitrary and Ω connected it follows easily that u takes the
constant value u(x̃) in Ω̄. In particular, the maximum is also attained on Γ .

30 3 Elliptic Equations
3.4 Fundamental Solutions. Green’s Function
Let u be a solution of the inhomogeneous equation
(3.12) Au = f in Rd,
where A is as in (3.1), with b = 0. Multiplying by ϕ ∈ C∞0 (R
d), integrating
over Rd, and integrating by parts twice, we obtain
(3.13) (u, Aϕ) = (f, ϕ) =
∫
Rd
f (x) ϕ(x) dx, ∀ϕ ∈ C∞0 (R
d).
We say that U is a fundamental solution of (3.12) if U is smooth for x ̸= 0, has
a singularity at x = 0 such that U ∈ L1(B), where B = {x ∈ R
d : |x| < 1}, and (3.14) |DαU (x)| ≤ Cα|x| 2−d−|α| for |α| ≠ 0, and if (3.15) (U, Aϕ) = ϕ(0), ∀ϕ ∈ C∞0 (R d). This means that, in the sense of weak derivative (see (A.21)), AU = δ, where δ is Dirac’s delta, defined in Problem A.9. We now use the fundamental solution to construct a solution to (3.12). Theorem 3.4. If U is a fundamental solution of (3.12) and if f ∈ C10 (R d), then u(x) = (U ∗ f )(x) = ∫ Rd U (x − y)f (y) dy is a solution of (3.12). Proof. We have, by (3.15), ∫ Rd U (x − y)Aϕ(x) dx = ∫ Rd U (z)Aϕ(z + y) dz = (U, Aϕ(· + y)) = ϕ(y). Hence, if u = U ∗ f , then, by changing the order of integration, (u, Aϕ) = ∫ Rd ∫ Rd U (x − y)f (y) dy Aϕ(x) dx = ∫ Rd ∫ Rd U (x − y)Aϕ(x) dx f (y) dy = ∫ Rd ϕ(y) f (y) dy = (f, ϕ). (3.16) 3.4 Fundamental Solutions. Green’s Function 31 Since f ∈ C10 it follows that u ∈ C 2 because with Di = ∂/∂xi we have DiDj u(x) = (DiU ∗ Dj f )(x) (cf. App. A.3) and DiU ∈ L1(R d) and Dj f ∈ C0(R d). Thus we may integrate by parts in (3.16) to obtain (cf. (3.13)) (Au − f, ϕ) = 0, ∀ϕ ∈ C∞0 (R d), from which we conclude that Au = f . ⊓. In the next theorem we determine fundamental solutions for Poisson’s equation in two and three dimensions. Theorem 3.5. Let U (x) = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ − 1 2π log |x|, when d = 2, 1 4π|x| , when d = 3. Then U is a fundamental solution for Poisson’s equation (3.3). Proof. We carry out the proof for d = 2; the proof for d = 3 is similar. By differentiation we find, for x ̸= 0, − ∂U ∂xj = 1 2π xj |x|2 , − ∂2U ∂x2j = 1 2π |x|2 − 2x2j |x|4 , so that, in particular, −∆U = 0 for x ̸= 0. Similarly, (3.14) holds. Let ϕ ∈ C∞0 (R 2). We have by Green’s formula, with n = x/|x|, ∫ |x|>ϵ
U (−∆ϕ) dx =
∫
|x|>ϵ
(−∆U )ϕ dx −
∫
|x|=ϵ
(
ϕ
∂U
∂n
−
∂ϕ
∂n
U
)
ds.
Note that n points inwards here. The first term on the right side vanishes.
Further, since
∂U
∂n
=
x1
|x|
∂U
∂x1
+
x2
|x|
∂U
∂x2
=
1
2π
1
|x|
=
1
2πϵ
, for |x| = ϵ,
we have
∫
|x|=ϵ
ϕ
∂U
∂n
ds =
1
2πϵ
∫
|x|=ϵ
ϕ ds → ϕ(0), as ϵ → 0.
Also,
∣
∣
∣
∫
|x|=ϵ
∂ϕ
∂n
U ds
∣
∣
∣ =
∣
∣
∣
1
2π
log(ϵ)
∫
|x|=ϵ
∂ϕ
∂n
ds
∣
∣
∣ ≤ ϵ| log(ϵ)| ∥∇ϕ∥C → 0, as ϵ → 0.
Hence
(U, (−∆)ϕ) = lim
ϵ→0
∫
|x|>ϵ
U (x)(−∆)ϕ(x) dx = ϕ(0).
⊓.

32 3 Elliptic Equations
We may now construct a Green’s function for the boundary value problem
(3.17) −∆u = f in Ω, with u = 0 on Γ,
namely a function G(x, y) defined for x, y ∈ Ω such that the solution of (3.17)
may be represented as
(3.18) u(x) =
∫
Ω
G(x, y)f (y) dy.
Let
(3.19) G(x, y) = U (x − y) − vy(x),
where U is the fundamental solution for −∆ from Theorem 3.5 and, for fixed
y ∈ Ω, let vy be the solution of
−∆xvy(x) = 0 in Ω, with vy(x) = U (x − y) on Γ.
In the next section we shall show that this problem has a solution. The
Green’s function thus has the singularity of the fundamental solution and
vanishes for x ∈ Γ , and it is easily seen that the function defined by (3.18)
is therefore a solution of (3.17). It is also the only solution, because we have
already proved uniqueness in Sect. 3.2. Note that G(x, y) consists of a singular
part, U (x − y) with a singularity at x = y, and a smooth part, vy(x).
3.5 Variational Formulation of the Dirichlet Problem
We first consider the Dirichlet problem with homogeneous boundary condi-
tions
(3.20) Au := −∇ ·
(
a∇u
)
+ b · ∇u + cu = f in Ω, with u = 0 on Γ,
where the coefficients a, b, and c are smooth functions in Ω̄ which satisfy
(3.21) a(x) ≥ a0 > 0, c(x) −
1
2
∇ · b(x) ≥ 0, for x ∈ Ω,
and where f is a given function. In the classical formulation of this problem
one looks for a function u ∈ C2 = C2(Ω̄) which satisfies (3.20). In this section
we shall reformulate (3.20) in variational form and seek a solution in the
larger class H10 . In some cases it is then possible to prove such regularity for
this solution that it is indeed a classical solution.
Assuming first that u is a solution in C2, we multiply (3.20) by v ∈ C10
and integrate over Ω. By Green’s formula and since v = 0 on Γ , we find that
(3.22)
∫
Ω
f v dx =
∫
Ω
Au v dx =
∫
Ω
(a ∇u·∇v +b·∇u v +c u v) dx ∀v ∈ C10 ,

3.5 Variational Formulation of the Dirichlet Problem 33
and then also, since C10 is dense in H
1
0 ,
(3.23)
∫
Ω
(
a ∇u · ∇v + b · ∇u v + c u v
)
dx =
∫
Ω
f v dx, ∀v ∈ H10 .
The variational problem corresponding to (3.20) is thus to find u ∈ H10 such
that (3.23) holds. It will be shown below, by means of the Lax-Milgram
lemma, that this problem admits a unique solution for f ∈ L2. We say that
this solution is a weak or variational solution of (3.20).
We have thus seen that a classical solution is also a weak solution. Con-
versely, suppose that u ∈ H10 is a weak solution, i.e., u satisfies (3.23). If in
addition we know that u ∈ C2, then by Green’s formula we have from (3.23)
∫
Ω
f v dx =
∫
Ω
(
a ∇u · ∇v + b · ∇u v + c u v
)
dx =
∫
Ω
A u v dx, ∀v ∈ H10 ,
i.e., ∫
Ω
(Au − f )v dx = 0, ∀v ∈ H10 .
If f ∈ C we have Au − f ∈ C, and therefore this relation implies
Au(x) − f (x) = 0, ∀x ∈ Ω.
Because u ∈ H10 we also have u = 0 on Γ , and it follows that u is a classical
solution of (3.20). A weak solution which is smooth enough is thus also a
classical solution. However, depending on the data f and the domain Ω, a
weak solution may or may not be smooth enough to be a classical solution and
the weak formulation (3.23) therefore really constitutes an extension of the
classical formulation. Note that the weak formulation (3.23) is meaningful
for any f ∈ L2, so that, e.g., f may be discontinuous, while the classical
formulation (3.20) requires f to be continuous. If f ∈ L2 and u ∈ H
2 ∩ H10
satisfies (3.20), then we say that u is a strong solution. Clearly, a classical
solution is also a strong solution, and a strong solution is a weak solution.
Further a weak solution that belongs to H2 is a strong solution. We shall
return below to the problem of the regularity of weak solutions.
We are now ready to show the existence of a weak solution. We use our
standard notation from Sect. 1.2.
Theorem 3.6. Assume that (3.21) holds and let f ∈ L2. Then the bound-
ary value problem (3.20) admits a unique weak solution, i.e., there exists a
unique u ∈ H10 which satisfies (3.23). Moreover, there exists a constant C
independent of f such that
(3.24) |u|1 ≤ C∥f∥.
Proof. We apply the Lax-Milgram lemma, Theorem A.3, in the Hilbert space
V = H10 equipped with the norm | · |1, and with

34 3 Elliptic Equations
(3.25) a(v, w) =
∫
Ω
(
a∇v ·∇w + b·∇v w + c v w
)
dx and L(v) =
∫
Ω
f v dx.
Clearly the bilinear form a(·, ·) is bounded in H10 and it is coercive when
(3.21) holds, since
a(v, v) =
∫
Ω
(
a|∇v|2 + (c − 1
2
∇ · b)|v|2
)
dx ≥ a0|v|
2
1, ∀v ∈ H
1
0 .
Further L(·) is a bounded linear functional on H10 , since by Poincaré’s in-
equality, Theorem A.6,
|L(v)| ≤ ∥f∥ ∥v∥ ≤ C∥f∥ |v|1.
This implies that ∥L∥V ∗ ≤ C∥f∥ and the statement of the theorem thus
follows directly from Theorem A.3. ⊓.
We observe that when b = 0, (3.21) reduces to (3.2), and the bilinear
form a(·, ·) is an inner product on H10 . The theorem can then be proved by
means of the Riesz representation theorem. In this case Theorem A.2 shows
that the weak solution of (3.20) may also be characterized as follows:
Theorem 3.7. (Dirichlet’s principle.) Assume that (3.2) holds and that b =
0. Let f ∈ L2 and u ∈ H
1
0 be the solution of (3.23), and set
(3.26) F (v) = 1
2
∫
Ω
(
a|∇v|2 + c v2
)
dx −
∫
Ω
f v dx.
Then F (u) ≤ F (v) for all v ∈ H10 , with equality only for v = u.
Remark 3.1. If (3.20) is considered, e.g., to be a model of an elastic membrane
fixed at its boundary, then F (v) as defined by (3.26) is the potential energy
associated with the deflection v; the first term in F (v) corresponds to the
internal elastic energy and the second term is a load potential (analogous
interpretations can be made for other problems in mechanics and physics
that are modeled by (3.20)). Dirichlet’s principle in this case corresponds to
the Principle of Minimum Potential Energy in mechanics and (3.23) to the
Principle of Virtual Work.
We now consider the boundary value problem with inhomogeneous bound-
ary condition,
Au = f in Ω, with u = g on Γ,(3.27)
where we assume that f ∈ L2 and g ∈ L2(Γ ). The weak formulation of this
problem is then to find u ∈ H1 such that, with a(·, ·) and L(·) as in (3.25),
(3.28) a(u, v) = L(v), ∀v ∈ H10 , with γ u = g,

3.6 A Neumann Problem 35
where γ : H1 → L2(Γ ) is the trace operator, cf. Theorem A.4. For the
existence of a solution, we assume that the given function g on Γ is the trace
of some function u0 ∈ H
1, i.e., g = γu0. Setting w = u − u0, we then seek
w ∈ H10 satisfying
(3.29) a(w, v) = L(v) − a(u0, v), ∀v ∈ H
1
0 .
The right hand side is a bounded linear functional on H10 and hence it follows
by the Lax-Milgram lemma that there exists a unique w ∈ H10 satisfying
(3.29). Clearly, u = u0 + w satisfies (3.28) and γu = g. This solution is
unique, for if (3.27) had two weak solutions u1, u2 with the same data f, g,
then their difference u1 − u2 ∈ H
1
0 would be a weak solution of (3.20) with
f = 0, and hence the stability estimate (3.24) would imply u1 − u2 = 0, i.e.,
u1 = u2. Hence, (3.27) has a unique weak solution. In particular, the solution
u is independent of the choice of extension u0 of the boundary values g.
When b = 0, the weak solution u ∈ H1 can equivalently be characterized
as the unique solution of the minimization problem
inf
v∈H1
γv=g
(
1
2
∫
Ω
(
a|∇v|2 + c v2
)
dx −
∫
Ω
f v dx
)
.
3.6 A Neumann Problem
We now consider the Neumann problem
(3.30) Au := −∇ ·
(
a∇u
)
+ cu = f in Ω, with
∂u
∂n
= 0 on Γ,
where we now in addition to (3.2) require c(x) ≥ c0 > 0 in Ω, and where
f ∈ L2. (The case c = 0 is discussed in Problem 3.9.) For a variational
formulation of (3.30) we multiply the differential equation in (3.30) by v ∈ C1
(note that we do not require v to satisfy any boundary conditions), and
integrate over Ω using Green’s formula, to obtain
∫
Ω
f v dx =
∫
Ω
Au v dx = −
∫
Γ
a
∂u
∂n
v ds +
∫
Ω
(
a∇u · ∇v + c uv
)
dx,
so that since ∂u/∂n = 0 on Γ ,
(3.31)
∫
Ω
(
a∇u · ∇v + c uv
)
dx =
∫
Ω
f v dx, ∀v ∈ C1.
Conversely, if u ∈ C2 satisfies (3.31), then by Green’s formula we have
(3.32)
∫
Ω
(
Au − f
)
v dx +
∫
Γ
a
∂u
∂n
v ds = 0, ∀v ∈ C1.

36 3 Elliptic Equations
If we first let v vary only over C10 , we see that u must satisfy the differen-
tial equation in (3.30). Thus, the first term on the left-hand side of (3.32)
vanishes, and by varying v on Γ , we see that u also satisfies the boundary
condition in (3.30).
We are thus led to the following variational formulation of (3.30): Find
u ∈ H1 such that
(3.33) a(u, v) = L(v), ∀v ∈ H1,
where a(·, ·) and L(·) are as in (3.25) with b = 0.
We have seen that if u is a classical solution of (3.30), then u satisfies
(3.33). Conversely, if u satisfies (3.33) and in addition u ∈ C2, then u is a
classical solution of (3.30).
By the Riesz representation theorem we have this time the following ex-
istence, uniqueness, and stability result. Note that since c(x) ≥ c0 > 0 the
bilinear form a(·, ·) is an inner product on H1.
Theorem 3.8. If f ∈ L2, then the Neumann problem (3.30) admits a unique
weak solution, i.e., there is a unique function u ∈ H1 that satisfies (3.33).
Moreover,
∥u∥1 ≤ C∥f∥.
Remark 3.2. Note that the Neumann boundary condition ∂u/∂n = 0 on Γ is
not enforced explicitly in the variational formulation (3.33); the function u is
just required to belong to H1. The boundary condition is implicitly contained
in (3.33), since the test function v may be an arbitrary function in H1. Such
a boundary condition, which does not have to be enforced explicitly, is called
a natural boundary condition. In contrast, a boundary condition, such as the
Dirichlet condition u = g on Γ , which is imposed explicitly as part of the
variational formulation, is said to be an essential boundary condition.
Remark 3.3. The problem
(3.34) Au = f in Ω, with a
∂u
∂n
= g on Γ,
where f ∈ L2(Ω) and g ∈ L2(Γ ) can be given the variational formulation:
Find u ∈ H1 such that
(3.35) a(u, v) = L(v), ∀v ∈ H1,
where a(·, ·) is as in (3.25) with b = 0 and
L(v) =
∫
Ω
f v dx +
∫
Γ
gv ds.
By the Cauchy-Schwarz inequality and the trace inequality (Theorem A.4)
we have

3.7 Regularity 37
|L(v)| ≤ ∥f∥ ∥v∥ + ∥g∥L2(Γ )∥v∥L2(Γ ) ≤
(
∥f∥ + C∥g∥L2(Γ )
)
∥v∥1,
and thus L(·) is a bounded linear form on H1. The Riesz representation
theorem therefore yields the existence and uniqueness of a function u ∈ H1
satisfying (3.35). See also Problem 3.7.
3.7 Regularity
We have learned in Theorem 3.6 that for any f ∈ L2 the Dirichlet problem
(3.20) has unique weak solution u ∈ H10 . It can be proved that if Γ is smooth,
or if Γ is a convex polygon, then, in fact, u ∈ H2, and there is a constant C
independent of f such that
∥u∥2 ≤ C∥f∥.
Since f = Au, this may also be expressed as
(3.36) ∥u∥2 ≤ C∥Au∥, ∀u ∈ H
2
∩ H10 .
Note that, when applied with, e.g., A = −∆, this inequality means that it is
possible to estimate the L2-norm of all second order derivatives of a function
u, which vanishes on Γ , in terms of the L2-norm of the special combination
of second derivates of u given by the Laplacian −∆. We refer to Problem 3.10
for an example, with Ω neither smooth nor convex, for which the regularity
estimate (3.36) does not hold.
The inequality (3.36) shows that u and its first and second order deriva-
tives depend continuously on f in the sense that if u1 and u2 satisfy
−Aui = fi in Ω, with ui = 0 on Γ, for i = 1, 2,
then ( ∑
|α|≤2
∥Dαu1 − D
αu2∥
2
)1/2
≤ C∥f1 − f2∥.
If Γ is smooth, then (3.36) can be generalized as follows. For any integer
k ≥ 0 there is a constant C independent of f such that if u is the weak
solution of (3.20) with f ∈ Hk, then u ∈ Hk+2 ∩ H10 and
(3.37) ∥u∥k+2 ≤ C∥f∥k.
In particular, in view of Sobolev’s inequality, Theorem A.5, this implies that
if k > d/2, then u ∈ C2 and thus u is also a classical solution of (3.20).
When Γ is a polygon the situation is not so favorable. In fact, if A = −∆
and Ω ⊂ R2 has a corner with interior angle ω, then using polar coordinates
(r, ϕ) centered at the corner, with ϕ = 0 corresponding to one of the edges,
one can show that the solution of (3.20) behaves as u(r, ϕ) = c rβ sin(βϕ) near

38 3 Elliptic Equations
the corner, with β = π/ω. For such a function to have Hk-regularity near the
corner, it is necessary that (∂/∂r)ku(r, ϕ) ∈ L2(Ω0), where Ω0 ⊂ Ω contains
a neighborhood of the corner under consideration, but no other corners. But
this requires that
(
β(β − 1) · · · (β − k + 1)
)2
∫ b
0
r2(β−k)r dr < ∞ for b sufficiently small, or that 2(β − k) + 1 ≥ −1 (note that β − k + 1 = 0 when 2(β − k) + 1 = −1). This in turn means that ω ≤ π/(k − 1). For k = 2 all angles thus have to be ≤ π, i.e., Ω has to be convex. For k = 3 all angles have to be ≤ π/2, which is a serious restriction. We refer to Problem 3.10 for an example that illustrates this. 3.8 Problems Problem 3.1. Give a variational formulation and prove the existence and uniqueness of a weak solution of the Dirichlet problem − d∑ j,k=1 ∂ ∂xj ( ajk ∂u ∂xk ) + a0u = f in Ω, with u = 0 on Γ, where ajk(x) and a0(x) are functions in C(Ω̄) such that a0(x) ≥ 0 and the matrix (ajk(x)) is symmetric and uniformly positive definite in Ω, so that ajk(x) = akj (x) and d∑ j,k=1 ajk(x)ξj ξk ≥ κ d∑ j=1 ξ2j with κ > 0, for ξ ∈ R
d, x ∈ Ω.
Problem 3.2. Show that if u satisfies −∆u = f in Ω, u = 0 on Γ , where
f ∈ L2, then p = ∇u is the solution to the minimization problem
inf
q∈Hf
1
2
∫
Ω
|q|2 dx,
where
Hf =
{
q = (q1, . . . , qd) : qi ∈ L2, −∇ · q = f in Ω
}
.
Problem 3.3. Consider two bounded domains Ω1 and Ω2 with a common
boundary S and let Γi = ∂Ωi \ S, where ∂Ωi is the boundary of Ωi, i = 1, 2,
see Fig. 3.1.
Give a variational formulation of the following problem: Find ui defined
in Ωi, i = 1, 2, such that

3.8 Problems 39
Γ1
Γ2
SΩ1
Ω2
Fig. 3.1. Domain with interface.
−a1∆u1 = f1 in Ω1, −a2∆u2 = f2 in Ω2,
u1 = 0 on Γ1, u2 = 0 on Γ2,
and
u1 = u2, a1
∂u1
∂n
= a2
∂u2
∂n
on S,
where fi ∈ L2(Ωi), ai > 0 is a constant, for i = 1, 2, and n is a unit normal
to S. Prove existence and uniqueness of a solution. Give an interpretation
from physics.
Problem 3.4. Prove Friedrichs’ inequality
∥v∥L2(Ω) ≤ C
(
∥∇v∥2L2(Ω) + ∥v∥
2
L2(Γ )
)1
2
, for v ∈ C1,
where Ω is a bounded domain in Rd with boundary Γ . Hint: Integrate by
parts in the identity
∫
Ω
v2 dx =
∫
Ω
v2∆φ dx, where φ(x) = 1
2d
|x|2.
Problem 3.5. Prove
∥v∥ ≤ C
(
∥∇v∥2 +
(∫
Ω
v dx
)2)1
2
, for v ∈ C1,
where Ω is the unit square in R2. The inequality holds also when Ω
is a bounded domain in Rd. Hint: v(x) = v(y) +
∫ x1
y1
D1v(s, x2) ds +∫ x2
y2
D2v(y1, s) ds.
Problem 3.6. Give a variational formulation of the problem
−∆u = f in Ω, with
∂u
∂n
+ u = g on Γ,
where f ∈ L2(Ω) and g ∈ L2(Γ ). Prove existence and uniqueness of a weak
solution. Give an interpretation of the boundary condition in connection with
some problem in mechanics or physics. Hint: See Problem 3.4.

40 3 Elliptic Equations
Problem 3.7. Prove the stability estimate
∥u∥H1(Ω) ≤ C
(
∥f∥L2(Ω) + ∥g∥L2(Γ )
)
for the solution of (3.34).
Problem 3.8. Give a variational formulation of the problem
−∇ ·
(
a∇u
)
+ cu = f in Ω, with a
∂u
∂n
+ h(u − g) = k on Γ,
where f ∈ L2(Ω), g, k ∈ L2(Γ ), and the coefficients a, c, h are smooth and
such that
a(x) ≥ a0 > 0, c(x) ≥ 0 for x ∈ Ω, h(x) ≥ h0 > 0 for x ∈ Γ.
Prove existence and uniqueness of a weak solution. Prove the stability esti-
mate
∥u∥H1(Ω) ≤ C
(
∥f∥L2(Ω) + ∥k∥L2(Γ ) + ∥g∥L2(Γ )
)
.
Hint: Use Problem 3.4.
Problem 3.9. Consider the Neumann problem
(3.38) −∆u = f in Ω, with
∂u
∂n
= 0 on Γ.
(a) Assume that f ∈ L2(Ω) and show that the condition
∫
Ω
f dx = 0.
is necessary for the existence of a solution.
(b) Notice that if u satisfies (3.38), then so does u + c for any constant c.
To obtain uniqueness, we add the extra condition
∫
Ω
u dx = 0,
requiring the mean value of u to be zero. Give this problem a variational
formulation using the space
V =
{
v ∈ H1(Ω) :
∫
Ω
v dx = 0
}
.
Prove that there is a unique weak solution. Hint: See Problem 3.5.
(c) Show that if the weak solution u ∈ V belongs to H2, then it solves
−∆u = f −
∫
Ω
f dx in Ω, with
∂u
∂n
= 0 on Γ.

3.8 Problems 41
Problem 3.10. Let Ω be a sector with angle ω = π/β:
Ω =
{
(r, ϕ) : 0 < r < 1, 0 < ϕ < π/β } , where r, ϕ are polar coordinates in the plane. Let v(r, ϕ) = rβ sin(βϕ). Verify that v is harmonic, i.e., ∆v = 0, by computing ∆v = 1 r ∂ ∂r ( r ∂v ∂r ) + 1 r2 ∂2v ∂ϕ2 . (This also follows immediately by noting that v is the imaginary part of the complex analytic function zβ .) Set u(r, ϕ) = (1 − r2)v(r, ϕ). Then u = 0 on Γ . Show that u satisfies −∆u = f with f = 4(1 + β)v. Hence f ∈ H1(Ω). Then compute ∥∂2u/∂r2∥L2(Ω) and conclude that u ̸∈ H 2(Ω) if β < 1, i.e., if Ω is non-convex or ω > π. Show in a similar way that u ̸∈ H3(Ω) if ω > π/2.
Hint: The most singular term in urr is β(β − 1)r
β−2 sin(βϕ).
Problem 3.11. (Elliptic regularity for a rectangle.) Assume that Ω ⊂ R2 is
a rectangle and that u is a smooth function with u = 0 on Γ . Prove that
|u|2 = ∥∆u∥.
Use this to prove (3.36) for A = −∆.
Hint: Recall that
|u|22 =
∫
Ω
((∂2u
∂x21
)2
+ 2
( ∂2u
∂x1∂x2
)2
+
(∂2u
∂x22
)2)
dx
and integrate by parts in
∫
Ω
(
∂2u
∂x1∂x2
)2
dx. Then recall the definition ∥u∥2 =
(
∥u∥2 + |u|21 + |u|
2
2
)1/2
and prove that ∥u∥ ≤ C|u|1 and |u|1 ≤
(
∥u∥ |u|2
)1/2
.
For arbitrary convex domains one can prove |u|2 ≤ ∥∆u∥ by a slightly
more complicated argument based on the same idea.
Problem 3.12. Replace the boundary condition in Problem 3.11 by the Neu-
mann condition ∂u/∂n = 0 on Γ . Prove that |u|2 = ∥∆u∥.
Problem 3.13. (Stability with respect to the coefficient.) Let ui, i = 1, 2,
be the weak solutions of the problems
−∇ ·
(
ai∇ui
)
= f in Ω, with ui = 0 on Γ,
where Ω ⊂ Rd is a domain with appropriately smooth boundary Γ , f ∈
L2(Ω), and the coefficients ai(x) are smooth and such that
ai(x) ≥ a0 > 0 for x ∈ Ω.
Prove the stability estimate
|u1 − u2|1 ≤
C
a20
∥a1 − a2∥C∥f∥.

4 Finite Difference Methods for Elliptic
Equations
The early development of numerical analysis of partial differential equations
was dominated by finite difference methods. In such a method an approximate
solution is sought at the points of a finite grid of points, and the approxi-
mation of the differential equation is accomplished by replacing derivatives
by appropriate difference quotients. This reduces the differential equation
problem to a finite linear system of algebraic equations. In this chapter we
illustrate this for a two-point boundary value problem in one dimension and
for the Dirichlet problem for Poisson’s equation in the plane. The analysis
is based on discrete versions of the maximum principles of the previous two
chapters.
4.1 A Two-Point Boundary Value Problem
We consider the two-point boundary value problem
(4.1)
Au := − au′′ + bu′ + cu = f in Ω = (0, 1),
u(0) = u0, u(1) = u1,
where the coefficients a = a(x), b = b(x), and c = c(x) are smooth functions
satisfying a(x) > 0 and c(x) ≥ 0 in Ω̄, and where the function f = f (x) and
the numbers u0, u1 are given.
For the purpose of numerical solution of (4.1) we introduce M + 1 mesh-
points 0 = x0 < x1 < · · · < xM = 1 by setting xj = jh, j = 0, . . . , M , where h = 1/M . We denote the approximation of u(xj ) by Uj and define the following finite difference approximations of derivatives, ∂Uj = Uj+1 − Uj h , ∂̄Uj = Uj − Uj−1 h , ∂∂̄Uj = Uj+1 − 2Uj + Uj−1 h2 , ∂̂Uj = Uj+1 − Uj−1 2h . With the notation of Sect. 1.2 we have with Cj = Cj (Ω̄) (see Problem 4.1), (4.2) |∂∂̄u(xj ) − u ′′(xj )| ≤ Ch 2 |u|C4 , |∂̂u(xj ) − u ′(xj )| ≤ Ch 2 |u|C3 , for j = 1, . . . , M − 1. 44 4 Finite Difference Methods for Elliptic Equations Setting also aj = a(xj ), bj = b(xj ), cj = c(xj ), fj = f (xj ), we now define a finite difference approximation of (4.1) by (4.3) AhUj := − aj ∂∂̄Uj + bj ∂̂Uj + cj Uj = fj , for j = 1, . . . , M − 1, U0 = u0, UM = u1. The equation at the interior point xj may be written (4.4) (2aj + h 2cj )Uj − (aj − 1 2 hbj )Uj+1 − (aj + 1 2 hbj )Uj−1 = h 2fj . Our discrete problem (4.3) may thus be put in matrix form as (4.5) AU = g, where U = (U1, . . . , UM−1) T and where the first and last components of the vector g = (g1, . . . , gM−1) T contain contributions from the boundary values u0, u1 as well as f1 and fM−1, respectively. The (M − 1) × (M − 1) matrix A is tridiagonal and diagonally dominant for h sufficiently small, i.e., the sum of the absolute values of the off-diagonal elements in one row is bounded by the diagonal element in that row, see Problem 4.2. For our analysis we first show a discrete maximum principle similar to that in the continuous case, cf. Theorem 2.1. Lemma 4.1. Assume that h is so small that aj ± 1 2 hbj ≥ 0 and that U satisfies AhUj ≤ 0 (AhUj ≥ 0). (i) If c = 0, then max j Uj = max{U0, UM } ( min j Uj = min{U0, UM } ) , (ii) If c ≥ 0, then max j Uj ≤ max{U0, UM , 0} ( min j Uj ≥ min{U0, UM , 0} ) . Proof. (i) In view of (4.4) we have, since c = 0 and AhUj ≤ 0, Uj = aj − 1 2 hbj 2aj Uj+1 + aj + 1 2 hbj 2aj Uj−1 + h2 2aj AhUj ≤ aj − 1 2 hbj 2aj Uj+1 + aj + 1 2 hbj 2aj Uj−1. (4.6) Assume now that U has an interior maximum Uj . Then if either Uj+1 < Uj or Uj−1 < Uj this would contradict (4.6), since the coefficients on the right are nonnegative and add up to 1. Hence, Uj = Uj−1 = Uj+1 and the latter values are also maxima. Continuing in this way we conclude that if the maximum is attained in the interior, then U is constant, and the maximum is thus also attained at the endpoints. This proves (i). Case (ii) is treated in the same way as case (ii) of Theorem 2.1. The versions with minimum are shown by considering −Uj . ⊓. 4.1 A Two-Point Boundary Value Problem 45 In the same way as for the continuous problem the maximum principle leads to a stability estimate in the discrete maximum-norm, as we shall now demonstrate. We assume for simplicity that b = 0. In this chapter we shall write for mesh-functions, (4.7) |U|S = max xj ∈S |Uj|. Lemma 4.2. Let Ah be as in (4.3), with b = 0. Then we have, for any mesh- function U , |U|Ω̄ ≤ max{|U0|, |UM |} + C|AhU|Ω. The constant C depends on the coefficients of A but not on h or U . Proof. Let w(x) = x − x2 = 1 4 − (x − 1 2 )2 and Wj = w(xj ). Then, with a = minΩ̄ a(x), AhWj = 2aj + cj (xj − x 2 j ) ≥ 2a. Setting V ±j = ±Uj − (2a) −1|AhU|ΩWj , we have hence AhV ± j = ±AhUj − (2a) −1 |AhU|ΩAhWj ≤ 0, so that we may apply Lemma 4.1 (note that the condition on h required is automatically satisfied when b = 0). Since W0 = WM = 0 we obtain ±Uj − (2a) −1 |AhU|ΩWj = V ± j ≤ max{±U0, ±UM , 0} ≤ max{|U0|, |UM |}. Since Wj ≤ 1 4 this shows the lemma with C = (8a)−1, ⊓. Lemma 4.2 immediately shows the existence and uniqueness of the solu- tion of (4.3) when b = 0. For uniqueness it suffices to note that if AhU = 0 and U0 = UM = 0, then U = 0, and the uniqueness implies the existence of a solution, since we are in a finite dimensional situation. For the case when b ̸= 0 we refer to Problem 4.3. We are now ready for an error estimate, which again for simplicity we demonstrate for b = 0 only. Theorem 4.1. Let b = 0, and let U and u be the solutions of (4.3) and (4.1). Then |U − u|Ω ≤ Ch 2 ∥u∥C4 . Proof. We have for the error zj = Uj − u(xj ) at the interior mesh-points Ahzj = AhUj − Ahu(xj ) = fj − Ahu(xj ) = Au(xj ) − Ahu(xj ) =: τj . By (4.2) we have for the truncation error (4.8) |τj| = | − aj (u ′′(xj ) − ∂∂̄u(xj ))| ≤ Ch 2 ∥u∥C4 , so that the result follows by Lemma 4.2, since z0 = zM = 0. ⊓. 46 4 Finite Difference Methods for Elliptic Equations 4.2 Poisson’s Equation We consider the Dirichlet problem for Poisson’s equation, (4.9) −∆u = f in Ω, with u = 0 on Γ, where Ω is a domain in R2 with boundary Γ . To begin with we assume that Ω is a square, Ω = (0, 1) × (0, 1) = {x = (x1, x2), 0 < xl < 1, l = 1, 2}. To define a finite difference approximation we write j = (j1, j2), where j1, j2 are integers and consider the mesh-points xj = jh, with h = 1/M the mesh-width, and mesh-functions U , with Uj = U (xj ). With e1 = (1, 0), e2 = (0, 1) we use the difference quotients (4.10) ∂lUj = Uj+el − Uj h , ∂̄lUj = Uj − Uj−el h , ∂l∂̄lUj = Uj+el − 2Uj + Uj−el h2 , l = 1, 2. Setting fj = f (xj ) we then replace (4.9) by (4.11) −∆hUj := − ∂1∂̄1Uj − ∂2∂̄2Uj = fj , for xj ∈ Ω, Uj = 0, for xj ∈ Γ. The difference equation at the interior mesh-points in Ω may be written (4.12) 4Uj − Uj+e1 − Uj−e1 − Uj+e2 − Uj−e2 = h 2fj , for xj ∈ Ω, which is the famous 5-point approximation of Poisson’s equation. The prob- lem (4.11) may thus be written in matrix form as AU = g, where A is a symmetric (M − 1)2 × (M − 1)2 matrix whose elements are 4, −1, or 0, with 0 the most common occurrence, and Ū the vector of interior nodal values. We have the following discrete maximum principle. Lemma 4.3. If U is such that −∆hUj ≤ 0 (−∆hUj ≥ 0) for xj ∈ Ω, then U attains its maximum (minimum) for some xj ∈ Γ . Proof. We may write, at the interior mesh-points, Uj = Uj+e1 + Uj−e1 + Uj+e2 + Uj−e2 4 − 1 4 h2∆hUj , so that −∆hUj ≤ 0 implies Uj ≤ 1 4 (Uj+e1 + Uj−e1 + Uj+e2 + Uj−e2 ). If Uj is an interior maximum, then Uj ≥ 1 4 (Uj+e1 +Uj−e1 +Uj+e2 +Uj−e2 ). Therefore equality holds, and the maximum value is taken also at all the neighboring points xj±el , which are therefore also maximum points. Continuing in the same way we conclude that if the maximum is attained in the interior, then U is constant. This proves the lemma. ⊓. 4.2 Poisson’s Equation 47 As before the maximum principle implies a stability estimate. Using again the notation (4.7) we have the following. Lemma 4.4. With ∆h defined in (4.11) we have, for any mesh-function U , |U|Ω̄ ≤ |U|Γ + C|∆hU|Ω. Proof. The proof is analogous to that of Theorem 3.2. With x̄ = ( 1 2 , 1 2 ) and x = (x1, x2) we set w(x) = 1 2 − |x − x̄|2 = x1 + x2 − x 2 1 − x 2 2 and define the mesh-function Wj = w(xj ). Then Wj ≥ 0 in Ω and −∆hWj = 4. Setting V ±j = ±Uj − 1 4 |∆hU|ΩWj we conclude that −∆hV ± j = ∓∆hUj − |∆hU|Ω ≤ 0, and, since Wj ≥ 0 for xj ∈ Γ , it follows from Lemma 4.3 that V ± j ≤ |U|Γ . Since Wj ≤ 1 2 in Ω this implies our statement with C = 1/8. ⊓. In particular, Lemma 4.4 implies the uniqueness of the solution of (4.11), and hence also the existence of a solution. In the same way as for the two- point boundary value problem the lemma also implies an error estimate. Theorem 4.2. Let U and u be the solutions of (4.11) and (4.9). Then |U − u|Ω ≤ Ch 2 |u|C4 . Proof. The error zj = Uj − u(xj ) satisfies, at the interior mesh-points, −∆hzj = fj + ∆hu(xj ) = −∆u(xj ) + ∆hu(xj ) =: τj , where τ is the truncation error, which may easily be estimated as in (4.2) by (4.13) |τj| ≤ 2∑ l=1 ∣ ∣ ∣∂l∂̄lu(xj ) − ∂2u ∂x2l (xj ) ∣ ∣ ∣ ≤ Ch2|u|C4 . The result therefore follows by application of Lemma 4.4 to zj , since zj = 0 for xj ∈ Γ . ⊓. The above analysis uses the fact that all the neighbors of the interior mesh-points in Ω are either interior mesh-points or belong to Γ . In the case of a curved boundary this cannot be achieved. We shall briefly discuss such a situation. We assume for simplicity that Ω is a convex plane domain with a smooth boundary Γ . We denote by Ωh those interior mesh-points xj for which all four neighbors of xj are also in Ω̄. (For the above case of a square, Ωh simply consists of all interior mesh-points.) Let now ωh be the mesh-points in Ω that are not in Ωh. For each xj ∈ ωh we may then select a (not necessarily unique) neighbor xi ∈ Ωh ∪ ωh such that the horizontal or vertical line through xj 48 4 Finite Difference Methods for Elliptic Equations and xi cuts Γ at a point x̄j , which is not a mesh-point (see Fig. 4.1). For this xj ∈ ωh, we then define the linear interpolation operator (4.14) ℓhUj := Uj − αj Ui − (1 − αj )U (x̄j ), where αj = |xj − x̄j| h + |xj − x̄j| ≤ 1 2 . Denoting by Γh the points of Γ which are either neighbors of points in Ωh or points x̄j associated with points in ωh, we now pose the problem (4.15) −∆hUj = fj in Ωh, ℓhUj = 0 in ωh, and U = 0 on Γh. xi xj x̄j Fig. 4.1. Interpolation near the boundary. This time we have the following stability estimate. Lemma 4.5. With ∆h defined in (4.11) and ℓh in (4.14), we have, for any mesh-function U , |U|Ωh∪ωh ≤ 2 ( |U|Γh + |ℓhU|ωh + C|∆hU|Ωh ) . Proof. Similarly to the proof of Lemma 4.4 we obtain |U|Ωh ≤ |U|ωh∪Γh + C|∆hU|Ωh . Here, for xj ∈ ωh, we have Uj = ℓhUj + αj Ui + (1 − αj )U (x̄j ), with 0 ≤ αj ≤ 1 2 , and hence |U|ωh ≤ |ℓhU|ωh + 1 2 |U|Ωh∪ωh + |U|Γh . Together these estimates show |U|Ωh∪ωh ≤ |U|ωh∪Γh + C|∆hU|Ωh ≤ |ℓhU|ωh + 1 2 |U|Ωh∪ωh + |U|Γh + C|∆hU|Ωh , which completes the proof. ⊓. 4.3 Problems 49 Again this shows uniqueness and existence of a solution of (4.15). Note that in this case the corresponding matrix A is nonsymmetric as, for instance, the elements aij and aji corresponding to the points xi and xj in Fig. 4.1 are different. We conclude with the following error estimate. Theorem 4.3. Let U and u be the solutions of (4.15) and (4.9). Then |U − u|Ωh∪ωh ≤ Ch 2 ∥u∥C4 . Proof. As in the proof of Theorem 4.2 we consider zj = Uj − u(xj ), and now apply Lemma 4.5. The only new term is ℓhzj = −ℓhu(xj ) and hence |ℓhzj| ≤ Ch 2|u|C2 , which completes the proof. ⊓. The above method of interpolation near the boundary is attributed to L. Collatz. It is also possible to use a five point finite difference approximation of −∆ based on nonuniform spacing on ωh, the so-called Shortley-Weller approximation. This also yields an O(h2) error estimate. 4.3 Problems Problem 4.1. Prove (4.2) and (4.13) by means of Taylor’s formula. Problem 4.2. Derive (4.5) and show that the matrix A is tridiagonal and (row-wise) diagonally dominant, i.e., ∑ j ̸=i |aij| ≤ aii, if h is sufficiently small. Hint: assume aj ± 1 2 hbj ≥ 0. Problem 4.3. Show that the conclusion of Lemma 4.2 (and hence that of Theorem 4.1) holds also when b ̸= 0, if h is sufficiently small and if we have at our disposal a mesh-function W such that AhWj ≥ 1 for xj ∈ Ω and Wj ≥ 0 for xj ∈ Ω̄. Construct such a function. (Hint: use the function w(x) = e λ−eλx with λ suitably chosen.) Problem 4.4. (Computer exercise.) Consider the two-point boundary value problem −u′′ + u = 2x in (0, 1), with u(0) = u(1) = 0. Apply the finite difference method (4.3) with h = 1/10, 1/20. Find the exact solution and compute the maximum of the error at the mesh-points. Problem 4.5. (Computer exercise.) Consider the Dirichlet problem (4.9) with f (x) = sin(πx1) sin(πx2) + sin(πx1) sin(2πx2) in Ω = (0, 1) × (0, 1). Compute the approximate solution by the finite dif- ference method (4.11) with h = 1/10, 1/20, and find the error at (0.5, 0.5), using that the exact solution is u(x) = (2π2)−1 sin(πx1) sin(πx2) + (5π 2)−1 sin(πx1) sin(2πx2). 5 Finite Element Methods for Elliptic Equations Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the under- lying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite ele- ment method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori er- ror estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integra- tion, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method. 5.1 A Two-Point Boundary Value Problem We consider the special case b = 0 of the two-point boundary value problem treated in Sect. 2.3, (5.1) Au := −(au′)′ + cu = f in Ω := (0, 1), with u(0) = u(1) = 0, where a = a(x), c = c(x) are smooth functions with a(x) ≥ a0 > 0, c(x) ≥ 0
in Ω̄, and f ∈ L2 = L2(Ω). We recall that the variational formulation of this
problem is to find u ∈ H10 such that

52 5 Finite Element Methods for Elliptic Equations
(5.2) a(u, ϕ) = (f, ϕ), ∀ϕ ∈ H10 ,
where
a(v, w) =
∫
Ω
(av′w′ + cvw) dx and (f, v) =
∫
Ω
f v dx,
and that this problem has a unique solution u ∈ H2.
For the purpose of finding an approximate solution of (5.2) we introduce
a partition of Ω,
0 = x0 < x1 < · · · < xM = 1, and set hj = xj − xj−1, Kj = [xj−1, xj ], for j = 1, . . . , M, and h = max j hj . The discrete solution will be sought in the finite-dimensional space of func- tions Sh = { v ∈ C = C(Ω̄) : v linear on each Kj , v(0) = v(1) = 0 } . (By a linear function we understand a function of the form f (x) = αx + β; strictly speaking such a function is called an affine function when β ̸= 0.) It is easy to see that Sh ⊂ H 1 0 . The set {Φi} M−1 i=1 ⊂ Sh of hat functions defined by Φi(xj ) = { 1, if i = j, 0, if i ̸= j, see Fig. 5.1, is a basis for Sh, and any v ∈ Sh may be written as v(x) = M−1∑ i=1 viΦi(x), with vi = v(xi). We now pose the finite-dimensional problem to find uh ∈ Sh such that (5.3) a(uh, χ) = (f, χ), ∀χ ∈ Sh. In terms of the basis {Φi} M−1 i=1 we write uh(x) = ∑M−1 j=1 Uj Φj (x) and insert this into (5.3) to find that this equation is equivalent to (5.4) M−1∑ j=1 Uj a(Φj , Φi) = (f, Φi), for i = 1, . . . , M − 1. This linear system of equations may be expressed in matrix form as (5.5) A U = b, 5.1 A Two-Point Boundary Value Problem 53 xi−2 xi−1 xi xi+1 Φi−1 Φi Fig. 5.1. Hat functions. where U = (Ui), A = (aij ) is the stiffness matrix with elements aij = a(Φj , Φi), and b = (bi) the load vector with elements bi = (f, Φi). The matrix A is symmetric and positive definite, because for V = (Vi) and v(x) = ∑M−1 i=1 ViΦi(x) we have V TA V = M−1∑ i,j=1 Viaij Vj = a ( M−1∑ j=1 Vj Φj , M−1∑ i=1 ViΦi ) = a(v, v) ≥ a0∥v ′ ∥ 2, and hence V TAV = 0 implies v′ = 0, so that v is constant = 0 because v(0) = 0, and thus V = 0. It follows that (5.5), and therefore also (5.3), has a unique solution, which is the finite element solution of (5.1). The matrix A is tridiagonal since aij = 0 when xi and xj are not neighbors, i.e., when |i − j| ≥ 2, and the system (5.5) is therefore easy to solve, see App. B.1. We note that when Au = −u′′ and the meshsize is constant, i.e., when hj = h = 1/M for j = 1, . . . , M, then, with the notation of Sect. 4.1, the equation (5.4) may be written (5.6) −∂∂̄Uj = h −1(f, Φj ), j = 1, . . . , M − 1 (cf. Problem 5.2). The finite element method thus coincides with the finite difference equation (4.3), except that an average of f over (xj − h, xj + h) is now used instead of the point-values fj = f (xj ). The idea of replacing the space H10 in (5.2) by a finite-dimensional sub- space and to determine the coefficients of the corresponding approximate solution as in (5.4) is referred to as Galerkin’s method. The finite element method is thus Galerkin’s method, applied with a special choice of the finite- dimensional subspace, namely, in this case, the space of continuous, piecewise linear functions. The intervals Kj , together with the restriction of these func- tions to Kj , are then thought of as the finite elements. 54 5 Finite Element Methods for Elliptic Equations Before we analyze the error in the finite element solution uh, we discuss some approximation properties of the space Sh. We define the piecewise linear interpolant Ihv ∈ Sh of a function v ∈ C = C(Ω̄) with v(0) = v(1) = 0 by Ihv(xj ) = v(xj ), j = 1, . . . , M − 1. Recall that H10 ⊂ C in one dimension by Sobolev’s inequality, Theorem A.5, so that Ihv is defined for v ∈ H 1 0 . It may be shown, which we leave as an exercise, see Problem 5.1, that, with ∥v∥Kj = ∥v∥L2(Kj ) and |v|2,Kj = |v|H2(Kj ), (5.7) ∥Ihv − v∥Kj ≤ Ch 2 j |v|2,Kj and (5.8) ∥(Ihv − v) ′ ∥Kj ≤ Chj|v|2,Kj . It follows that ∥Ihv − v∥ = ( M∑ j=1 ∥Ihv − v∥ 2 Kj )1/2 ≤ ( M∑ j=1 C2h4j |v| 2 2,Kj )1/2 ≤ Ch2∥v∥2, ∀v ∈ H 2, (5.9) and similarly (5.10) ∥(Ihv − v) ′ ∥ ≤ Ch∥v∥2, for v ∈ H 2. We now turn to the task of estimating the error in the finite element approximation uh defined by (5.3). Since a(·, ·) is symmetric positive definite, it is an inner product on H10 , and the corresponding norm is the energy norm (5.11) ∥v∥a = a(v, v) 1/2 = (∫ 1 0 ( a(v′)2 + cv2 ) dx )1/2 . Theorem 5.1. Let uh and u be the solutions of (5.3) and (5.2). Then (5.12) ∥uh − u∥a = min χ∈Sh ∥χ − u∥a, and (5.13) ∥u′h − u ′ ∥ ≤ Ch∥u∥2. Proof. Since Sh ⊂ H 1 0 we may take ϕ = χ ∈ Sh in (5.2) and subtract it from (5.3) to obtain (5.14) a(uh − u, χ) = 0, ∀χ ∈ Sh. This equation means that the finite element solution uh may be described as the orthogonal projection of the exact solution u onto Sh with respect to 5.1 A Two-Point Boundary Value Problem 55 the inner product a(·, ·). This also immediately implies that uh is the best approximation of u in Sh with respect to the energy norm, and hence that (5.12) holds. This can be seen directly as follows: Using (5.14) we have, for any χ ∈ Sh, ∥uh − u∥ 2 a = a(uh − u, uh − u) = a(uh − u, χ − u) ≤ ∥uh − u∥a ∥χ − u∥a, which shows (5.12) after cancellation of a factor ∥uh − u∥a. By our assump- tions we have, with C independent of h, √ a0∥v ′ ∥ ≤ ∥v∥a ≤ C∥v ′ ∥, for v ∈ H10 , where the first inequality is obvious by (5.11) and the second follows from (2.17). Hence, (5.12) implies (5.15) ∥(uh − u) ′ ∥ ≤ C∥uh − u∥a ≤ C min χ∈Sh ∥(χ − u)′∥. Taking χ = Ihu and using the interpolation error bound in (5.10), we obtain (5.13), and the proof is complete. ⊓. Our next result concerns the L2-norm of the error. Theorem 5.2. Let uh and u be the solutions of (5.3) and (5.2). Then (5.16) ∥uh − u∥ ≤ Ch 2 ∥u∥2. Proof. We use a duality argument based on the auxiliary problem (5.17) Aφ = e in Ω, with φ(0) = φ(1) = 0, where e = uh − u. Its weak formulation is to find φ ∈ H10 such that (5.18) a(w, φ) = (w, e), ∀w ∈ H10 . We put the test function w on the left side, because (5.18) plays the role of the adjoint (or dual) problem to (5.2). Of course, this makes no difference here since a(·, ·) is symmetric, but is important in the case of a nonsymmetric differential operator A, see Problem 5.7. By the regularity estimate (2.22) we have (5.19) ∥φ∥2 ≤ C∥Aφ∥ = C∥e∥. Taking w = e in (5.18) and using (5.14) and (5.10), we therefore obtain ∥e∥2 = a(e, φ) = a(e, φ − Ihφ) ≤ C∥e ′ ∥ ∥(φ − Ihφ) ′ ∥ ≤ Ch∥e′∥ ∥φ∥2 ≤ Ch∥e ′ ∥ ∥e∥. Cancelling one factor ∥e∥ we see that we have gained one factor h over the error estimate for e′, (5.20) ∥e∥ ≤ Ch∥e′∥, and the proof may now be completed by using (5.13). ⊓. 56 5 Finite Element Methods for Elliptic Equations Remark 5.1. We note that the above error estimates contain the norm of the second order derivative, while the fourth order derivative was needed in the corresponding result for the finite difference method in Theorem 4.1. This is related to the fact that in the finite element method the load term f enters via averages rather than through point-values as in the finite difference scheme. This will be discussed further in Sect. 5.6 below. Remark 5.2. The solution of the very special equation (5.6) agrees with the nodal values of the exact solution of the corresponding two-point boundary value problem. In fact, with u = u(x) the exact solution, we have by Taylor’s formula ∂∂̄u(xj ) = h −2 ∫ xj xj−1 (y − xj−1)u ′′(y) dy + h−2 ∫ xj+1 xj (xj+1 − y)u ′′(y) dy = h−1(u′′, Φj ) = −h −1(f, Φj ). Thus the finite element solution uh is identical to the interpolant Ihu of the exact solution. For a discussion of this based on the Green’s function, see Problem 5.4. In the above analysis we could have considered a more general finite el- ement space, consisting of piecewise polynomials of degree r − 1, where r is an integer ≥ 2, with the above piecewise linear case included for r = 2, thus with Sh = { v ∈ C : v ∈ Πr−1 on each Kj , v(0) = v(1) = 0 } , where Πk denotes the space of polynomials of degree ≤ k. In addition to the hat functions above we may then associate with each interval Ki the basis functions Φij ∈ Πr−1 on Ki for j = 1, . . . , r − 2, and vanishing outside Ki, defined by Φij (xi,l) = { 1, if j = l, 0, if j ̸= l, where xi,l = xi−1 + hi l r − 1 , l = 0, . . . , r − 1. Using also these additional nodal points in the definition of the interpolant Ihv one may show the local estimates ∥Ihv − v∥Kj ≤ Ch r j ∥v (r) ∥Kj and ∥(Ihv − v) ′ ∥Kj ≤ Ch r−1 j ∥v (r) ∥Kj , and consequently the global estimates (5.21) ∥Ihv − v∥ ≤ Ch r ∥v∥r and ∥(Ihv − v) ′ ∥ ≤ Chr−1∥v∥r, ∀v ∈ H r. For the finite element solution one then obtains, in the same way as above, (5.22) ∥uh − u∥ ≤ Ch r ∥u∥r and ∥u ′ h − u ′ ∥ ≤ Chr−1∥u∥r. These inequalities thus require v, u ∈ Hr. Using that the interpolant Ihv is well defined for v ∈ H10 , one can show that they also hold with r replaced by any s with 1 ≤ s ≤ r. The case s = 2 of the second estimate in (5.21) is needed in the proof of the O(hr) estimate in (5.22) by duality. 5.2 A Model Problem in the Plane 57 xi−1xi−1 xixixi,1 xi,1 xi,2 Φi−1Φi−1 ΦiΦiΦi1 Φi1 Φi2 Fig. 5.2. Global basis functions for r = 3 and 4. 5.2 A Model Problem in the Plane Let now Ω be a polygonal domain in R2, i.e., a domain whose boundary Γ is a polygon, and consider the simple model problem (5.23) Au := −∇ · ( a∇u ) = f in Ω, with u = 0 on Γ. We assume that the coefficient a = a(x) is smooth with a(x) ≥ a0 > 0 in Ω̄
and that f ∈ L2.
We recall from Sect. 3.5 that the variational formulation of (5.23) is to
find u ∈ H10 such that
(5.24) a(u, v) = (f, v), ∀v ∈ H10 ,
where
a(v, w) =
∫
Ω
a∇v · ∇w dx and (f, v) =
∫
Ω
f v dx,
and that this problem has a unique solution in H10 . Moreover, if Ω is assumed
to be convex, then the regularity estimate in (3.36) implies that u ∈ H2 and
(5.25) ∥u∥2 ≤ C∥f∥.
Our discussion of the approximation of (5.23) follows similar lines as
for the two-point boundary value problem above. This time we divide the
polygonal domain Ω into triangles. More precisely, let Th = {K} be a set of
closed triangles K, a triangulation of Ω, such that
Ω̄ =
⋃
K∈Th
K, hK = diam(K), h = max
K∈Th
hK .
The vertices P of the triangles K ∈ Th are called the nodes of the triangula-
tion Th. We require that the intersection of any two triangles of Th is either
empty, a node, or a common edge, and that no node is located in the interior
of an edge of Th, see Fig. 5.3.

58 5 Finite Element Methods for Elliptic Equations
Fig. 5.3. Invalid (left) and valid (right) triangulation.
With the triangulation Th we associate the function space Sh consisting
of continuous, piecewise linear functions on Th, vanishing on Γ , i.e.,
Sh =
{
v ∈ C(Ω̄) : v linear in K for each K ∈ Th, v = 0 on Γ
}
.
Using our above assumptions on Th it is not difficult to verify that Sh ⊂ H
1
0 .
Let {Pi}
Mh
i=1 be the set of interior nodes, i.e., those that do not lie on Γ . A
function in Sh is then uniquely determined by its values at the Pj , and the
set of pyramid functions {Φi}
Mh
i=1 ⊂ Sh, defined by
Φi(Pj ) =
{
1, if i = j,
0, if i ̸= j,
forms a basis for Sh. If v ∈ Sh we thus have v(x) =
∑Mh
i=1 viΦi(x), where the
vi = v(Pi) are the nodal values of v. It follows that Sh is a finite-dimensional
subspace of the Hilbert space H10 .
The finite element approximation of the problem (5.24) is then to find
uh ∈ Sh such that
(5.26) a(uh, χ) = (f, χ), ∀χ ∈ Sh.
Using the basis {Φi}
Mh
i=1 we write uh(x) =
∑Mh
i=1 UiΦi(x), which, inserted into
(5.26), gives a linear system of equations for the determination of the Uj ,
(5.27)
Mh∑
j=1
Uj a(Φj , Φi) = (f, Φi), i = 1, . . . , Mh,
This may be written in matrix form as AU = b, where U = (Ui), A = (aij )
is the stiffness matrix with elements aij = a(Φj , Φi), and b = (bi) the load
vector with elements bi = (f, Φi). The matrix A is symmetric and positive
definite as in Sect. 5.1, so that (5.27) and hence (5.26) has a unique solution
in Sh. Moreover, the matrix A is large and sparse if the mesh is fine, i.e., a
large portion of its elements are zero because each Φi vanishes except in the
union of the triangles that contain the node Pi, so that aij = a(Φj , Φi) = 0

5.2 A Model Problem in the Plane 59
unless Pi and Pj are neighbors. This property is important for the efficient
solution of the linear system, cf. App. B. This time the finite elements are
the triangles K ∈ Th together with the restrictions to the K of the functions
in Sh.
More generally, given the triangulation Th we can take Sh to be the func-
tions on Ω, which reduce to polynomials of degree r − 1 on the triangles
K ∈ Th, where r is a fixed integer ≥ 2. It may be shown that such a function
χ is uniquely determined by its values at a certain finite number of nodes in
each K, which can be chosen in different ways. In the case r = 3, i.e., when
Sh consists of piecewise quadratic functions, these points may be taken to be
the vertices of Th together with the midpoints of the edges in Th, altogether
6 points for each K ∈ Th. For piecewise cubics, i.e., when r = 4, we may take
the vertices of Th, two interior points on each edge of Th, and the barycenter
of each K ∈ Th, thus using 10 points for each K ∈ Th, see Fig. 5.4. Note
that a polynomial in two variables of second and third degree is determined
uniquely by the values of 6 and 10 coefficients, respectively, and to deter-
mine these we need to require this number of linear conditions, or degrees of
freedom, as they are referred to in this context.
The finite element space Sh thus defined is still a finite dimensional sub-
space of H10 , and one basis function Φj ∈ Sh may be associated with each
of the nodes described. The finite element problem (5.26) and its matrix
formulation (5.5) remain of the same form as before.
Fig. 5.4. Triangles with 6 and 10 nodes.
If the boundary Γ of Ω is not a polygon but a smooth curve, then a
triangulation of the above type will not fit Ω exactly. If Ω is convex it is
possible to choose the triangulation in such a way that the union Ωh of the
triangles still approximates Ω, by choosing the boundary vertices of Ωh on
Γ , so that the set Ω \ Ωh of points in Ω not covered by the triangulation
has a width of order O(h2), see Fig. 5.5. Defining the functions in Sh to
vanish on Ω \ Ωh, a finite element solution uh can be defined as above. It
turns out that for Sh consisting of piecewise linear functions nothing is lost
by this extension, see Sect. 5.3 below, but for piecewise polynomials of higher
degree the situation is not so favorable. Various modifications of the methods

60 5 Finite Element Methods for Elliptic Equations
have then been devised to deal with the approximation near Γ . We shall not
go into details but remark that at any rate a triangulation provides a more
flexible way of approximating a domain Ω than is possible with the square
mesh used in the finite difference method, and that this is a useful property
of the finite element method.
Fig. 5.5. Smooth convex domain with triangulation.
In the sequel we take the point of view that we consider not only one
triangulation Th and associated function space Sh, but a whole family of
triangulations {Th}0 2, the contributions in
(5.36) and (5.37) are the best one can expect, and therefore the first inequality
in (5.34) holds with r = 2 and 3, but the second only for r = 2.
We close with a remark about the orthogonal projection Ph = PSh of the
Hilbert space L2 onto the finite-dimensional subspace Sh, which is defined
by
(5.38) (Phv − v, χ) = 0, ∀χ ∈ Sh, v ∈ L2.
Since Phv is the best approximation of v in Sh with respect to the L2-norm,
and hence by the above, in the case of a polygonal domain,
(5.39) ∥Phv − v∥ ≤ ∥Ihv − v∥ ≤ Ch
r
∥v∥r, ∀v ∈ H
r
∩ H10 .
Here we use the notation Hr ∩ H10 for the space of functions that belong to
Hr and vanish on Γ . The requirement that v ∈ Hr ∩H10 is a rather strong one
and not normally satisfied for solutions of our elliptic problem when r > 2
because of the singularities at the corners of the domain, which we discussed
at the end of Sect. 3.7. For a convex domain with smooth boundary the
regularity is not a problem, but without further modification of the method
near the boundary, we only know (5.39) to hold for r = 2 and 3.

5.4 Error Estimates 63
5.4 Error Estimates
We return to the task of estimating the error in the finite element approxi-
mation uh of the solution u of our Dirichlet problem. Since the bilinear form
a(·, ·) is an inner product in H10 it is natural to use the energy norm
∥v∥a = a(v, v)
1/2 =
(∫
Ω
a|∇v|2 dx
)1/2
.
Theorem 5.3. Let uh and u be the solutions of (5.26) and (5.24). Then
(5.40) ∥uh − u∥a = min
χ∈Sh
∥χ − u∥a,
and
(5.41) |uh − u|1 ≤ Ch∥u∥2.
Proof. Since Sh ⊂ H
1
0 we may take v = χ ∈ Sh in (5.24) and subtract it from
(5.26) to obtain
(5.42) a(uh − u, χ) = 0, ∀χ ∈ Sh,
which means that uh is the orthogonal projection of u onto Sh with respect
to the inner product a(·, ·). The equality (5.40) hence follows in the same way
as (5.12). In view of our assumptions on a we have, with C and c independent
of h,
(5.43) c|v|1 ≤ ∥v∥a ≤ C|v|1.
Hence, (5.40) implies
(5.44) |uh − u|1 ≤ C min
χ∈Sh
|χ − u|1.
Taking χ = Ihu and using the interpolation error bound in (5.32), this proves
(5.41). ⊓.
For the analogous result in the case of a nonsymmetric elliptic operator,
see Problems 5.6 and 5.7.
The equality (5.40) means that uh is the best, or optimal, approximation
of u in Sh with respect to the energy norm, and (5.44) shows that it is an
almost best, or quasi-optimal, approximation in the standard Sobolev norm
in H10 . Note that the energy norm is a weighted norm in H
1
0 ; in order to take
full advantage of the best approximation property (5.40) one would need to
prove a weighted variant of the interpolation error bound (5.32). This can be
done but we will not pursue it here. Of course, these norms coincide when
a = 1.

64 5 Finite Element Methods for Elliptic Equations
For (5.41) to be of interest it is necessary that u ∈ H2. In the case that
Ω is convex we know from Sect. 3.7 that such regularity follows from f ∈ L2,
and that (5.25) holds. From (5.41) we therefore conclude that
|uh − u|1 ≤ Ch∥f∥,
where the constant is the product of those in (5.41) and (5.25). If Ω is noncon-
vex, then the solution u will generally have such singularities at the corners
of Γ that will make (5.25) invalid, and this will result in lower order of con-
vergence. Note that (5.40) still holds in this case.
Our next result concerns the L2-norm of the error. Here we need the
regularity estimate (5.25) and therefore assume that Ω is convex.
Theorem 5.4. Let Ω be convex and let uh and u be the solutions of (5.26)
and (5.24). Then
(5.45) ∥uh − u∥ ≤ Ch
2
∥u∥2.
Proof. The proof proceeds as for the two-point boundary value problem in
Theorem 5.2 by duality, using the auxiliary problem
(5.46) Aφ = e in Ω, with φ = 0 on Γ, where e = uh − u.
We have as in (5.25)
(5.47) ∥φ∥2 ≤ C∥e∥,
and this is used as in Theorem 5.2 to show
(5.48) ∥e∥ ≤ Ch|e|1.
By Theorem 5.3 this completes the proof. ⊓.
The previous theorems show the same error bounds for uh as for the inter-
polant Ihu in (5.31) and (5.32), except that the constants may be different.
Note that we have only used (5.32) and not (5.31) in the proofs.
Let Rh : H
1
0 → Sh be the orthogonal projection with respect to the energy
inner product, so that
(5.49) a(Rhv − v, χ) = 0, ∀χ ∈ Sh, v ∈ H
1
0 .
The operator Rh is called the Ritz projection (or elliptic projection). It follows
from (5.42) that the finite element solution uh is exactly the Ritz projection
of the exact solution u of (5.24), i.e., uh = Rhu. Our previous error estimates
for the finite element solution may be expressed as follows in terms of the op-
erator Rh, which will be convenient when we discuss parabolic finite element
problems later.

5.4 Error Estimates 65
Theorem 5.5. Let Ω be convex. Then we have, for s = 1, 2,
∥Rhv − v∥ ≤ Ch
s
∥v∥s, |Rhv − v|1 ≤ Ch
s−1
∥v∥s, ∀v ∈ H
s
∩ H10 .
Proof. The case s = 2 is contained in Theorems 5.3 and 5.4. For the case
s = 1 we first note that since Rh is the orthogonal projection with respect to
a(·, ·), we have ∥Rhv∥a ≤ ∥v∥a. Hence |Rhv|1 ≤ C|v|1 and |Rhv−v|1 ≤ C|v|1.
Finally, using (5.48) we obtain
∥Rhv − v∥ ≤ Ch|Rhv − v|1 ≤ Ch∥v∥1,
which completes the proof. ⊓.
Formally the above error analysis extends immediately to finite elements
of higher order r > 2. In the argument in Theorem 5.4 we simply use the
second interpolation error estimate in (5.34) instead of (5.32), together with
the case s = 2 of (5.35). We then find, for 2 ≤ s ≤ r,
(5.50) ∥Rhv − v∥ ≤ Ch
s
∥v∥s, |Rhv − v|1 ≤ Ch
s−1
∥v∥s, ∀v ∈ H
s
∩ H10 .
These estimates thus show a reduced convergence rate O(hs) if v ∈ Hs with
s < r. As we pointed out at the end of Sect. 5.3, the regularity assumption v ∈ Hr with r > 2 is somewhat unrealistic for solutions of our elliptic problem in a
polygonal domain. For a domain Ω with a smooth boundary Γ the regularity
is not a problem but special considerations for handling the boundary layer
Ω \ Ωh are then needed to attain high accuracy.
Because the variational formulation of our discrete problem is based on
L2 inner products, the most natural error estimates are also expressed in
such L2 based norms, and therefore measure certain averages of the error. It
is, of course, also of interest to derive error bounds in the maximum-norm,
which express uniform error bounds over Ω. We first note that the error in
the interpolant introduced above satisfies
∥Ihv − v∥C(K) ≤ Ch
2
K∥v∥C2(K), ∀K ∈ Th,
and thus, also in the case of a smooth boundary Γ , since then ∥v∥C(Ω\Ωh) ≤
Ch2∥v∥C1 , we have
(5.51) ∥Ihv − v∥C ≤ Ch
2
∥v∥C2 .
Under the additional assumption that the family of triangulations {Th} is
quasi-uniform, i.e., that
(5.52) hK ≥ ch
for some positive c independent of h, it is also possible, but not easy, to show
that, for our elliptic problem we have

66 5 Finite Element Methods for Elliptic Equations
(5.53) ∥uh − u∥C ≤ Ch
2 log(1/h)∥u∥C2 , for h small,
see Problem 5.4. Compared to the L2-norm estimate of Theorem 5.4 this
estimate contains an additional factor log(1/h), which is not present in the
interpolation error estimate (5.51), and it may be shown that this factor
cannot be removed.
5.5 An A Posteriori Error Estimate
The error bounds of the previous section contain norms of the exact, unkown,
solution. Using the regularity estimate (5.25) these error bounds may also be
expressed in terms of the data f of (5.23), and, if the constants entering
are known, stringent bounds for the error are obtained. However, as we have
seen in Sect. 5.3, these bounds could be pessimistic, particularly when the
triangulations are far from uniform, in which case, for instance, the inequality
(5.32) could be very crude. These estimates involving h = maxK hK should
therefore be interpreted as asymptotic estimates, showing the rate of con-
vergence of the error as h → 0. Thus, for example, Theorem 5.4 shows that
∥uh − u∥ = O(h
2) as h → 0 if u ∈ H2.
Since these bounds do not depend on the computed solution, they are
often referred to as a priori bounds; they may be stated before the computa-
tion has been carried out. In the next theorem we shall give an example of
an a posteriori error estimate, which is expressed in terms of the computed
solution and the data.
Theorem 5.6. Assume that Ω is a convex polygonal domain in the plane.
Let uh and u be the solutions of (5.26) and (5.24), respectively. Then
∥uh − u∥ ≤ C
( ∑
K∈Th
R2K
)1/2
,
where
RK = h
2
K∥Auh − f∥K + h
3/2
K ∥a[n · ∇uh]∥∂K\Γ ,
and [n · ∇uh] denotes the jump across ∂K in the normal derivative n · ∇uh.
Proof. We use the duality argument from the proof of Theorem 5.4. Set
e = uh−u and let φ be the solution of (5.46). Then, with (v, w)K =
∫
K
v w dx,
∥v∥K = ∥v∥L2(K), and |v|2,K = |v|H2(K),
∥e∥2 = a(e, φ) = a(uh − u, φ) = a(uh, φ) − (f, φ)
=
∑
K
(
(a∇uh, ∇φ)K − (f, φ)K
)
=
∑
K
(
(Auh − f, φ)K + (an · ∇uh, φ)∂K
)
=
∑
K
(
(Auh − f, φ)K −
1
2
(a[n · ∇uh], φ)∂K\Γ
)
,

5.6 Numerical Integration 67
where the factor 1/2 in the last term appears because the term occurs twice
in the sum. Since a(e, χ) = 0 for χ ∈ Sh, we may replace φ in the above by
φ − χ to obtain
∥e∥2 = |a(e, φ − χ)|
≤
∑
K
(
∥Auh − f∥K ∥φ − χ∥K +
1
2
∥a[n · ∇uh]∥∂K\Γ ∥φ − χ∥∂K\Γ
)
.
We now choose χ = Ihφ and recall (5.29), (5.30), and also the scaled trace
inequality, obtained by transformation of the trace inequality (A.26) from a
reference triangle K̂ of unit size to the small triangle K, see Problem A.15,
(5.54) ∥w∥∂K ≤ C
(
h
−1/2
K ∥w∥K + h
1/2
K ∥∇w∥K
)
.
Hence we obtain
(5.55) ∥φ − Ihφ∥∂K ≤ Ch
3/2
K |φ|2,K ,
and, in view of the regularity estimate (5.47), we may conclude
∥e∥2 = a(e, φ) ≤ C
∑
K
RK|φ|2,K ≤ C
(∑
K
R2K
)1/2(∑
K
|φ|22,K
)1/2
≤ C
(∑
K
R2K
)1/2
∥φ∥2 ≤ C
(∑
K
R2K
)1/2
∥e∥,
which completes the proof. ⊓.
If A = −∆, i.e., if a = 1, then Auh = 0 in K, and since n·∇uh is constant
along ∂K, we have RK = h
2
K
(
∥f∥K +
∣
∣[n·∇uh]|∂K\Γ
∣
∣), so that the computed
solution only enters in the second term. The a posteriori error estimate does
not by itself imply that the error converges to zero with h, but this follows
from the O(h2) a priori error estimate as shown before.
The a posteriori error estimate also suggests an approach to adaptive
error control, namely to refine the mesh by subdividing those triangles K
for which RK is large compared to some tolerance. We will not go into the
details.
5.6 Numerical Integration
An important feature of the finite element method is that the equations
(5.27) can be generated automatically by a computer program. This proce-
dure, which is called assembly of the equations, is based on an elementwise
computation of the stiffness matrix and the load vector,

68 5 Finite Element Methods for Elliptic Equations
(5.56) a(Φj , Φi) =
∑
K∈Th
∫
K
a∇Φj · ∇Φi dx, (f, Φi) =
∑
K∈Th
∫
K
f Φi dx.
In practice, the integrals in these sums are seldom computed exactly even
if analytical expressions for a and f are available. Instead they are approxi-
mated by numerical integration through a quadrature formula of the form
(5.57)
∫
K
φ dx ≈ qK (φ) :=
L∑
l=1
ωl,K φ(bl,K ).
The numbers ωl,K are called the weights and the points bl,K the nodes of the
quadrature formula.
If the equations are assembled by numerical integration, then instead of
(5.26) we solve a modified finite element problem, which is to find uh ∈ Sh
such that
(5.58) ah(uh, χ) = (f, χ)h, ∀χ ∈ Sh,
where
(5.59) ah(v, w) =
∑
K∈Th
qK (a∇v · ∇w), (f, w)h =
∑
K∈Th
qK (f w).
The quadrature formula qK in (5.57) should be chosen in such a way that
the error in uh is of the same order as in the original finite element solution.
An example of such a quadrature formula is the barycentric quadrature rule
(5.60) qK (φ) = |K|φ(PK ), where |K| = area(K), PK =
1
3
3∑
l=1
Pl,K ,
with Pl,K and PK the vertices and the barycenter of the triangle K. This
quadrature rule is exact for linear functions, i.e.,
(5.61)
∫
K
φ dx = |K|φ(PK ), ∀φ ∈ Π1.
This implies that the rule is accurate of order 2 so that (Problem 5.13)
(5.62)
∣
∣
∣qK (φ) −
∫
K
φ dx
∣
∣
∣ ≤ Ch2K|φ|W 21 (K),
where, with Dij = ∂
2/∂xi∂xj ,
|v|W 2
1
(M) =
2∑
i,j=1
∥Dij v∥L1(M ), ∥v∥L1(M) =
∫
M
|v| dx.
Hence, the global quadrature error is bounded by

5.6 Numerical Integration 69
(5.63)
∣
∣
∣
∑
K∈Th
qK (φ) −
∫
Ω
φ dx
∣
∣
∣ ≤ Ch2
∑
K∈Th
|φ|W 2
1
(K).
From (5.61) we conclude that ah(uh, χ) and (f, χ)h are exact, for example,
when a and f are constant.
Another example of a quadrature formula, which is exact for linear func-
tions, is provided by the nodal quadrature rule, see Problem 5.15,
(5.64) qK (φ) =
1
3
|K|
3∑
l=1
φ(Pl,K ).
In the following lemma we collect the properties of ah(·, ·) and (·, ·)h that
we need in order to prove an error estimate for the modified problem (5.58).
Lemma 5.1. If ah(·, ·) and (·, ·)h in (5.59) are computed by the quadrature
formula (5.60) or (5.64), then
(5.65) a0|χ|
2
1 ≤ ah(χ, χ) ≤ C|χ|
2
1, ∀χ ∈ Sh,
and
|ah(ψ, χ) − a(ψ, χ)| ≤ Ch
2
∥a∥C2 |ψ|1 |χ|1, ∀ψ, χ ∈ Sh,(5.66)
|(f, χ)h − (f, χ)| ≤ Ch
2
∥f∥2 |χ|1, ∀χ ∈ Sh.(5.67)
Proof. We carry out the proof for the quadrature rule (5.60); the proof for
(5.64) is analogous.
Since ∇χ is constant on K and a0 ≤ a(x) ≤ C, we have
ah(χ, χ) =
∑
K
a(PK )|∇χ(PK )|
2
|K| ≥ a0
∑
K
∫
K
|∇χ|2 dx = a0|χ|
2
1.
The estimate from above is derived in the same way, which shows (5.65).
Using (5.63) with φ = a∇ψ · ∇χ we get
|ah(ψ, χ) − a(ψ, χ)| ≤ Ch
2
∑
K
|a∇ψ · ∇χ|W 2
1
(K).
To bound the right hand side, we have for ψ, χ ∈ Sh,
∥Dij (a∇ψ · ∇χ)∥L1(K) = ∥(Dij a)∇ψ · ∇χ∥L1(K) ≤ ∥a∥C2 ∥∇ψ∥K ∥∇χ∥K .
Invoking the Cauchy-Schwarz inequality for sums, we conclude
∑
K
|a∇ψ · ∇χ|W 2
1
(K) ≤ C∥a∥C2 |ψ|1 |χ|1,
which proves (5.66). Similarly, since Dij χ = 0 on K, we have

70 5 Finite Element Methods for Elliptic Equations
∥Dij (f χ)∥L1(K) = ∥Dij f χ + Dif Dj χ + Dj f Diχ∥L1(K) ≤ C∥f∥2,K ∥χ∥1,K ,
so that ∑
K
|f χ|W 2
1
(K) ≤ C∥f∥2 ∥χ∥1 ≤ C∥f∥2 |χ|1,
which proves (5.67). ⊓.
The inequality (5.65) shows that the symmetric bilinear form ah(·, ·) is
an inner product on Sh and that the corresponding norm is equivalent to
| · |1, uniformly with respect to h. From (5.67) we deduce that the linear form
Lh(χ) = (f, χ)h is bounded on Sh with respect to | · |1, again uniformly with
respect to h, because
|(f, χ)h| ≤ |(f, χ)| + |(f, χ)h − (f, χ)|
≤ ∥f∥ ∥χ∥ + Ch2∥f∥2 |χ|1 ≤ C∥f∥2 |χ|1.
By the Riesz representation theorem we may therefore conclude that (5.58)
has a unique solution and that it satisfies the stability estimate
(5.68) |uh|1 ≤ C∥f∥2,
see Problem 5.14. This stability of the modified finite element problem is used
together with the consistency error bounds (5.66), (5.67) in the proof of the
following error estimate.
Theorem 5.7. Assume that ah(·, ·) and (·, ·)h in (5.59) are computed by the
quadrature formula (5.60) or (5.64). Let uh and u be the solutions of (5.58)
and (5.24), respectively. Then
(5.69) |uh − u|1 ≤ Ch∥u∥2 + Ch
2
(
∥a∥C2 ∥u∥2 + ∥f∥2
)
.
Proof. We write uh − u = (uh − Ihu) + (Ihu − u) = θ + ρ. Using (5.24) and
(5.58), we get, for any χ ∈ Sh,
ah(θ, χ) = ah(uh, χ) − ah(Ihu, χ) +
(
a(u, χ) − (f, χ)
)
−
(
ah(uh, χ) − (f, χ)h
)
+ a(Ihu, χ) − a(Ihu, χ)
= −a(ρ, χ) −
(
ah(Ihu, χ) − a(Ihu, χ)
)
+
(
(f, χ)h − (f, χ)
)
.
Since θ ∈ Sh we may choose χ = θ. In view of (5.65), the boundedness of the
bilinear form a(·, ·), and the error estimates (5.66) and (5.67), this implies
a0|θ|
2
1 ≤ ah(θ, θ) ≤
(
C|ρ|1 + Ch
2
∥a∥C2 |Ihu|1 + Ch
2
∥f∥2
)
|θ|1.
Hence,

5.7 A Mixed Finite Element Method 71
|uh − u|1 ≤ |θ|1 + |ρ|1 ≤ C|ρ|1 + Ch
2
(
∥a∥C2 |Ihu|1 + ∥f∥2
)
.
Using the interpolation error estimate (5.32), we have
|ρ|1 ≤ Ch∥u∥2, |Ihu|1 ≤ |u|1 + |ρ|1 ≤ |u|1 + Ch∥u∥2 ≤ C∥u∥2.
Together these estimates prove (5.69). ⊓.
The first term on the right side of (5.69) is (essentially) the same as in
(5.41), whereas the remaining terms estimate the effect of numerical integra-
tion. Note that this result requires more regularity than (5.41). For example,
we need f ∈ H2, which (at least formally) implies that u ∈ H4. This is con-
sistent with the result for the finite difference scheme in Theorem 4.2, where
u is required to have four derivatives, see also Remark 5.1. We may think
of the finite element method with numerical integration as a finite difference
scheme on a non-uniform mesh.
The original finite element method (5.26) is conforming in the sense that
Sh ⊂ H
1
0 and the forms a(·, ·) and L(·) = (f, ·) are the same as in the continu-
ous problem (5.24). In the modified finite element method (5.58) we still have
Sh ⊂ H
1
0 , but the forms are different. It is therefore called non-conforming.
Other non-conforming finite element methods violate the inclusion Sh ⊂ H
1
0 ,
for example, by the use of discontinuous piecewise polynomials. They can
sometimes be analyzed in a similar way. The argument used in the proof
of Theorem 5.7 is based on what is known as Strang’s first lemma in the
literature on non-conforming finite element methods.
5.7 A Mixed Finite Element Method
In some situations it is the flux, −a∇u, of the solution u that is of primary
interest. However, in the standard finite element method the derivatives, and
hence the flux, are approximated to lower order O(h) rather than the O(h2)
approximation of the solution. We shall now briefly describe a finite element
method for our model problem (5.23), which is based on a so called mixed
formulation of this problem, and which does not have this disadvantage. Here
the flux of the solution u is introduced as a separate dependent variable whose
approximation is sought in a different finite element space than the solution
itself. This may be done in such a way that the flux is approximated to the
same order of accuracy as u. For simplicity we assume that a = 1 in (5.23).
With σ = ∇u as a separate two-dimensional variable, this equation may then
be formulated as the system
(5.70)
−∇ · σ = f in Ω,
σ = ∇u in Ω,
u = 0 on ∂Ω.

72 5 Finite Element Methods for Elliptic Equations
With H = {ω = (ω1, ω2) ∈ L2 × L2 : ∇ · ω ∈ L2} we note that the solution
(u, σ) ∈ L2 × H also solves the variational problem
(5.71)
(∇ · σ, ϕ) + (f, ϕ) = 0, ∀ϕ ∈ L2,
(σ, ω) + (u, ∇ · ω) = 0, ∀ω ∈ H,
where the (·, ·) denotes the appropriate L2 inner products, and a smooth
solution of (5.71) satisfies (5.70). Setting L(v, µ) = 1
2
∥µ∥2 + (∇ · µ + f, v),
one may show that the solution (u, σ) of (5.70) can be characterized as the
saddle-point satisfying
(5.72) L(v, σ) ≤ L(u, σ) ≤ L(u, µ), ∀v ∈ L2, µ ∈ H
(see Problem 5.16), and the key to the existence of a solution is the inequality
(5.73) inf
v∈L
2
sup
µ∈H
(v, ∇ · µ)
∥v∥ ∥µ∥H
≥ c > 0, where ∥µ∥2H = ∥µ∥
2 + ∥∇ · µ∥2.
With Sh and Hh certain finite-dimensional subspaces of L2 and H we shall
consider the discrete analogue of (5.71), which is to find (uh, σh) ∈ Sh × Hh
such that
(5.74)
(∇ · σh, χ) + (f, χ) = 0, ∀χ ∈ Sh,
(σh, ψ) + (uh, ∇ · ψ) = 0, ∀ψ ∈ Hh.
As in the continuous case this problem is equivalent to the discrete analogue
of the saddle point problem (5.72), and in order for this discrete problem to
have a solution with the desired properties, the choice of spaces Sh ×Hh must
be such that the analogue of (5.73) holds, in this context referred to as the
Babuška-Brezzi inf-sup condition. More precisely,
(5.75) inf
v∈S
h
sup
µ∈Hh
(v, ∇ · µ)
∥v∥ ∥µ∥H
≥ c > 0,
must hold uniformly with respect to h.
An example of a pair of spaces which satisfy the inf-sup condition, intro-
duced by Raviart and Thomas, is as follows: With Th a quasi-uniform family
of triangulation of Ω, which we assume here to be polygonal, we set
Sh =
{
χ ∈ L2 : χ|K linear, ∀K ∈ Th
}
,
with no continuity required across inter-element boundaries. We also define
Hh =
{
ψ = (ψ1, ψ2) ∈ H : ψ|K ∈ H(K), ∀K ∈ Th
}
,
where H(K) denotes affine maps of quadratics on a reference triangle K̂ of
the form (l1(ξ) + αξ1(ξ1 + ξ2), l2(ξ) + βξ2(ξ1 + ξ2)), with l1(ξ), l2(ξ) linear,

5.8 Problems 73
α, β ∈ R. Since each of the functions lj (ξ) has three parameters we have
dim H(K) = 8. The space Hh thus consists of piecewise quadratics on the
triangulation Th, which are of the specific form implied by the definition of
H(K). As degrees of freedom for Hh one may use the values of ψ · n at two
points on each side of K (6 conditions) and in addition the mean-values of
ψ1 and ψ2 over K (2 conditions). We note that the condition ψ ∈ H in the
definition of Hh requires that ∇·ψ ∈ L2, which is equivalent to the continuity
of χ·n across inter-element boundaries. For the solutions of (5.74) and (5.70)
one may show
∥uh − u∥ ≤ Ch
2
∥u∥2 and ∥σh − σ∥ ≤ Ch
s
∥u∥s+1, s = 1, 2.
Thus, the flux σ is approximated to the same order O(h2) as u.
5.8 Problems
Problem 5.1. Prove (5.7) and (5.8).
Hint for (5.8): (Ihv)
′(x) − v′(x) = h−1j
∫
Kj
(
v′(y) − v′(x)
)
dy for x ∈ Kj .
Hint for (5.7): Let Q1v the polynomial of degree 1 obtained from Taylor’s
formula for v at xj−1. Note that Ih(Q1v) = Q1v and ∥Ihv∥C(Kj ) ≤ ∥v∥C(Kj ),
so that ∥Ihv − v∥C(Kj ) = ∥Ih(v − Q1v) + (Q1v − v)∥C(Kj ) ≤ 2∥v − Q1v∥C(Kj ).
Estimate the remainder: ∥v − Q1v∥C(Kj ) ≤ maxx∈Kj
∫
Kj
|x − y| |v′′(y)| dy.
Conclude ∥Ihv − v∥C(Kj ) ≤ 2hj
∫
Kj
|v′′(y)| dy, which implies (5.7). This proof
can be generalized to functions of two variables, see (5.29), the main difference
is that it is more difficult to estimate the remainder in Taylor’s formula.
Problem 5.2. Find the elements of the matrix A in (5.5) when hj = h =
constant.
Problem 5.3. Use the basis {Φi}
Mh
i=1 to show that (5.38) can be written in
matrix form as BV = b, where the matrix B (the so-called mass matrix) is
symmetric, positive definite, and sparse if Mh is large.
Problem 5.4. Consider the situation in Sect. 5.1 with a piecewise linear
finite element space Sh.
(a) Use the Green’s function in Theorem 2.3 to prove that uh = Ihu when a =
1, c = 0 in (5.1), cf. Remark 5.2. Hint: Use the results from Problem 2.4,
Problem 2.2 (a), and the fact that G(xj , ·) ∈ Sh if xj is a node.
(b) In the case of variable coefficients prove that
|uh(xj ) − u(xj )| ≤ Ch
2
∥u∥2.
Hint: Show that e(xj ) = a(e, G(xj , ·) − IhG(xj , ·)) and use an interpo-
lation error estimate on the intervals (0, xj ), (xj , 1), where G(xj , ·) is
smooth.

74 5 Finite Element Methods for Elliptic Equations
(c) Conclude that ∥uh − u∥C ≤ Ch
2∥u∥C2 , which is (5.53) in this simple
special case. Hint: ∥uh − Ihu∥C = maxj |uh(xj ) − u(xj )|.
This is the basic idea behind one approach to maximum-norm estimates for
elliptic problems in several variables. However, the stronger singularity of the
Green’s function, see Sect. 3.4, makes the analysis much more difficult.
Problem 5.5. Prove that, under the assumptions of Theorem 5.3,
|uh − u|1 ≤ C
(∑
K
h2K|u|
2
2,K
)1/2
.
Problem 5.6. (Galerkin’s method.) Let a(·, ·) and L(·) satisfy the assump-
tions of the Lax-Milgram lemma, i.e.,
|a(v, w)| ≤ C1∥v∥V ∥w∥V , ∀v, w ∈ V,
a(v, v) ≥ C2∥v∥
2
V , ∀v ∈ V,
|L(v)| ≤ C3∥v∥V , ∀v ∈ V.
Let u ∈ V be the solution of
a(u, v) = L(v), ∀v ∈ V.
Let Ṽ ⊂ V be a finite-dimensional subspace and let ũ ∈ Ṽ be determined by
Galerkin’s method:
a(ũ, v) = L(v), ∀v ∈ Ṽ .
Prove that (note that a(·, ·) may be non-symmetric)
∥ũ − u∥V ≤
C1
C2
min
χ∈Ṽ
∥χ − u∥V .
Prove that, if a(·, ·) is symmetric and ∥v∥a = a(v, v)
1/2, then
∥ũ − u∥a = min
χ∈Ṽ
∥χ − u∥a and ∥ũ − u∥V ≤
√
C1
C2
min
χ∈Ṽ
∥χ − u∥V .
Problem 5.7. Consider the problem
−∇ ·
(
a∇u
)
+ b · ∇u + cu = f in Ω, with u = 0 on Γ,
from Sect. 3.5. Note that the presence of the convection term b · ∇u makes
the bilinear form non-symmetric.
(a) Formulate a finite element method for this problem and prove an error
bound in the H1-norm. Hint: See Problem 5.6.

5.8 Problems 75
(b) Prove an error bound in the L2-norm. Hint: Modify the proof of Theo-
rem 5.4 by using the auxiliary problem
A
∗φ := −∇ · (a∇φ) − b · ∇φ + (c − ∇ · b)φ = e in Ω, φ = 0 on Γ,
instead of (5.46). The operator A∗ is the adjoint of A, defined by
(Av, w) = a(v, w) = (v, A∗w) for all v, w ∈ H2 ∩ H10 .
Problem 5.8. Formulate a finite element problem corresponding to the non-
homogeneous Dirichlet problem (3.27). Prove error estimates. Hint: With the
notation of Sect. 5.3, uh(x) =
∑Mh
j=1 Uj Φj (x) +
∑Nh
j=Mh+1
g(Pj )Φj (x).
Problem 5.9. Formulate a finite element problem corresponding to the Neu-
mann problem (3.30). Prove error estimates.
Problem 5.10. Formulate a finite element problem corresponding to the
nonhomogeneous Neumann problem (3.34). Prove error estimates.
Problem 5.11. Formulate a finite element problem corresponding to the
Robin problem in Problem 3.6. Prove error estimates.
Problem 5.12. The following important result is called the Bramble-Hilbert
lemma. Let F be a nonnegative functional on W mp = W
m
p (Ω), where Ω is a
bounded convex domain in Rd, and assume that
F (v + w) ≤ F (v) + F (w), ∀v, w ∈ W mp ,
F (v) ≤ C∥v∥W mp , ∀v ∈ W
m
p ,
F (v) = 0, ∀v ∈ Πm−1.
Then F is bounded with respect to the corresponding seminorm, i.e., there is
a constant C = C(Ω, m, p) such that
F (v) ≤ C|v|W mp , ∀v ∈ W
m
p .
(a) Prove the Bramble-Hilbert lemma for d = 1, Ω = (0, 1), and m = p = 2.
Hint: As in Problem 5.1 show that ∥v − Q1v∥2 ≤ C|v|2. Then F (v) ≤
F (v − Q1v) + F (Q1v) = F (v − Q1v) ≤ C∥v − Q1v∥2 ≤ C|v|2.
(b) Use the Bramble-Hilbert lemma to show (5.29) and (5.30). Hint: Do this
first for a fixed triangle K̂ of unit size and then make an affine trans-
formation of this triangle to a small triangle K, see Problem A.14. For
(5.29) choose W mp (Ω) = H
2(K̂), F (v) = ∥Ihv −v∥L2(K̂) and estimate the
nodal values by Sobolev’s inequality |v(P̂j )| ≤ C∥v∥H2(K̂).
Problem 5.13. Prove (5.62) by using the Bramble-Hilbert lemma with m =
2, p = 1.
Problem 5.14. Prove (5.68).

76 5 Finite Element Methods for Elliptic Equations
Problem 5.15. Prove an analogue of Lemma 5.1 for the nodal quadrature
formula (5.64).
Problem 5.16. Prove that any solution of (5.71) satisfies (5.72).
Problem 5.17. (Computer exercise.) Consider the same two-point bound-
ary value problem as in Problem 4.4. Apply the finite element method (5.3)
based on piecewise linear approximating functions on the same partition as
in Problem 4.4 with h = 1/10, 1/20. Find the exact solution and compute
the maximum of the error at the mesh-points.
Problem 5.18. (Computer exercise.) Consider the same boundary value
problem as in Problem 4.5. Solve it by the finite element method (5.26)
based on piecewise linear approximating functions on the same partition as
in Problem 4.5, divided into triangles by inserting a diagonal with positive
slope into each mesh-square, with h = 1/10, 1/20. Recall the exact solution
and compute the L2-norm of the error. Use the barycentric quadrature rule
to compute the stiffness matrix, the load vector, and the L2-norm.

6 The Elliptic Eigenvalue Problem
Eigenvalue problems are important in the mathematical analysis of partial
differential equations, and occur, e.g., in the modelling of vibrating mem-
branes and other applications. In our study of time-dependent partial differ-
ential equations it will be important to develop functions in eigenfunction
expansions, and we therefore discuss such expansions in Sect. 6.1 below. In
the following Sect. 6.2 we present some simple approaches and results for the
numerical solution of eigenvalue problems.
6.1 Eigenfunction Expansions
We shall first consider the eigenvalue problem corresponding to the symmetric
case of the two-point boundary value problem in Chapt. 2, to find a number
λ and a function ϕ, which is not identically zero, such that
(6.1) Aϕ := −(aϕ′)′ + cϕ = λϕ in Ω = (0, 1), with ϕ(0) = ϕ(1) = 0,
where a and c are smooth functions such that a(x) ≥ a0 > 0 and c(x) ≥ 0
on Ω̄. Such a number λ is called an eigenvalue and ϕ is the corresponding
eigenfunction.
Recall that the two-point boundary value problem
(6.2) Au = f in Ω, with u(0) = u(1) = 0,
may be written in weak form as: Find u ∈ H10 = H
1
0 (Ω) such that
a(u, v) = (f, v), ∀v ∈ H10 ,
where the bilinear form and the inner product are defined by
(6.3) a(u, v) =
∫ 1
0
(a u′v′ + c uv) dx and (u, v) =
∫ 1
0
uv dx,
respectively. With this notation the eigenvalue problem (6.1) may be stated:
Find a number λ and a function ϕ ∈ H10 , ϕ ̸= 0, such that
(6.4) a(ϕ, v) = λ (ϕ, v), ∀v ∈ H10 .

78 6 The Elliptic Eigenvalue Problem
We shall also consider the Dirichlet eigenvalue problem to find a number
λ and a function ϕ, not identically zero, such that
(6.5) −∆ϕ = λϕ in Ω, with ϕ = 0 on Γ,
where Ω is a bounded domain in Rd with smooth boundary Γ . Here the
associated Dirichlet boundary value problem is
(6.6) −∆u = f in Ω, with u = 0 on Γ,
or, in variational form, to find u ∈ H10 = H
1
0 (Ω) such that
a(u, v) = (f, v), ∀v ∈ H10 ,
where now
(6.7) a(u, v) =
∫
Ω
∇u · ∇v dx = (∇u, ∇v) and (u, v) =
∫
Ω
uv dx.
The variational form of the eigenvalue problem corresponding to (6.5) is again
to find a number λ and a function ϕ ∈ H10 , ϕ ̸= 0, such that (6.4) holds.
Recall that, if u is a solution of (6.6) and f ∈ Hk, then u ∈ Hk+2 ∩ H10 ,
see Sect. 3.7. Hence we may conclude at once that a possible eigenfunction is
smooth: Since ϕ ∈ L2, elliptic regularity implies that ϕ ∈ H
2 ∩ H10 , which in
turn shows ϕ ∈ H4 ∩ H10 , and so on. The corresponding observation applies
also to the simpler eigenvalue problem (6.1).
Both eigenvalue problems (6.1) and (6.6) thus have the variational for-
mulation (6.4), and this would also be the case if instead of the Laplacian
we used the more general elliptic operator Au = −∇ · (a∇u) + c u together
with suitable homogeneous boundary conditions of Dirichlet, Neumann, or
Robin type, as described in Chapt. 3. This would lead to the following more
general eigenvalue problem. Let H = L2 = L2(Ω) and V be a linear subspace
of H1 = H1(Ω), with Ω ⊂ Rd, and assume that the bilinear form a(·, ·) is
symmetric and coercive on V , i.e.,
(6.8) a(v, v) ≥ α∥v∥21, ∀v ∈ V, with α > 0.
Find ϕ ∈ V , ϕ ̸= 0, and a number λ such that
(6.9) a(ϕ, v) = λ(ϕ, v), ∀v ∈ V,
where (·, ·) denotes the inner product in H = L2. Note that a(·, ·) is an inner
product in V .
For simplicity we shall consider the concrete case of (6.5), with a(·, ·)
defined by (6.7), and we invite the reader to check that the theory that we
shall present actually applies, with minor notational changes, to the more
general eigenvalue problem (6.9).
We begin with some general simple properties of the eigenvalues and eigen-
functions.

6.1 Eigenfunction Expansions 79
Theorem 6.1. The eigenvalues of (6.5) are real and positive. Two eigen-
functions corresponding to different eigenvalues are orthogonal in L2 and
H10 .
Proof. Let λ be an eigenvalue and ϕ the corresponding eigenfunction. Then
λ∥ϕ∥2 = λ (ϕ, ϕ) = a(ϕ, ϕ),
which together with (6.8) implies that λ > 0. Let λ1 and λ2 be two different
eigenvalues and ϕ1 and ϕ2 the corresponding eigenfunctions. Then
λ1 (ϕ1, ϕ2) = a(ϕ1, ϕ2) = a(ϕ2, ϕ1) = λ2 (ϕ2, ϕ1) = λ2 (ϕ1, ϕ2),
so that
(λ1 − λ2)(ϕ1, ϕ2) = 0.
Since λ1 ̸= λ2 it follows that (ϕ1, ϕ2) = 0, and hence also a(ϕ1, ϕ2) = 0. ⊓.
As a first step we shall show that there exists an eigenvalue, in fact, a
smallest eigenvalue. This eigenvalue will be characterized by
(6.10) λ1 = inf
{
a(v, v) : v ∈ H10 , ∥v∥ = 1
}
, where a(v, v) = ∥∇v∥2.
The equality (6.10) may also be written (this is referred to as the Rayleigh-
Ritz characterization of the principal eigenvalue)
λ1 = inf
v ̸=0
∥∇v∥2
∥v∥2
,
which follows from ∥∇(αv)∥2 = α2∥∇v∥2. Since, for an arbitrary eigenvalue
λ and corresponding eigenfunction ϕ,
∥∇ϕ∥2 = a(ϕ, ϕ) = λ (ϕ, ϕ) = λ∥ϕ∥2,
we conclude that λ ≥ λ1, so that λ1 is a lower bound for the eigenvalues.
Theorem 6.2. The infimum in (6.10) is attained by a function ϕ1 ∈ H
1
0 .
This function is an eigenfunction of (6.5) and λ1 the corresponding eigen-
value.
Proof. We shall postpone the proof of the first statement of the theorem till
the end of this section and assume that the infimum is attained by ϕ1 ∈ H
1
0 ,
i.e., λ1 = ∥∇ϕ1∥
2 and ∥ϕ1∥ = 1. We now show that ϕ1 is an eigenfunction
of (6.5) corresponding to the eigenvalue λ1, that is,
(6.11) a(ϕ1, v) = λ1 (ϕ1, v), ∀v ∈ H
1
0 .
Note that, for α an arbitrary real number,

80 6 The Elliptic Eigenvalue Problem
a(ϕ1 + αv, ϕ1 + αv) = λ1 + 2α a(ϕ1, v) + α
2a(v, v)
and
∥ϕ1 + αv∥
2 = 1 + 2α (ϕ1, v) + α
2
∥v∥2.
Since the ratio of the two norms is bounded below by λ1 for all α, we have
λ1 + 2αa(ϕ1, v) + α
2a(v, v) ≥ λ1 + 2λ1α (ϕ1, v) + λ1α
2
∥v∥2,
or
2α
(
a(ϕ1, v) − λ1(ϕ1, v)
)
+ α2
(
a(v, v) − λ1∥v∥
2
)
≥ 0.
Suppose now that (6.11) does not hold, so that the coefficient of α is ̸= 0.
Then choosing |α| small and with a sign such that the first term is negative,
we have a contradiction. ⊓.
From Theorem 6.2 we thus know that at least one eigenfunction ϕ1 exists.
We now repeat the considerations above in the subspace V1 of V = H
1
0
consisting of functions which are orthogonal to ϕ1 with respect to (·, ·). Note
that these functions are then orthogonal to ϕ1 also with respect to a(·, ·)
since a(v, ϕ1) = λ1(v, ϕ1) = 0. We consider thus
λ2 = inf
{
a(v, v) : v ∈ V, ∥v∥ = 1, (v, ϕ1) = 0
}
= inf
{
∥∇v∥2 : v ∈ H10 , ∥v∥ = 1, (v, ϕ1) = 0
}
.
(6.12)
Clearly λ2 ≥ λ1, since the infimum here is taken over a smaller set of functions
v than in (6.10). In the same way as above we may show that the infimum
is attained and we call the minimizing function ϕ2 (we shall return to the
question of existence of ϕ2 later), which then satisfies
a(ϕ2, ϕ2) = ∥∇ϕ2∥
2 = λ2, ∥ϕ2∥ = 1, (ϕ1, ϕ2) = 0.
To show that ϕ2 is an eigenfunction, we first show, exactly as above that
a(ϕ2, v) = λ2 (ϕ2, v), for all v ∈ H
1
0 with (v, ϕ1) = 0.
To see that the equation holds for all v ∈ H10 and not just for those orthogonal
to ϕ1, we note that any v ∈ H
1
0 may be written as
v = αϕ1 + w, with α = (v, ϕ1) and (w, ϕ1) = 0.
It therefore remains only to show that a(ϕ2, ϕ1) = λ2 (ϕ2, ϕ1). But this fol-
lows at once since (ϕ2, ϕ1) = 0 and a(ϕ2, ϕ1) = 0.
Continuing in this way we find a nondecreasing sequence of eigenvalues
{λj}
∞
j=1 and a corresponding sequence of eigenfunctions {ϕj}
∞
j=1, which are
mutually orthogonal and have L2-norm 1, such that

6.1 Eigenfunction Expansions 81
λn = a(ϕn, ϕn)
= inf
{
a(v, v) : v ∈ H10 , ∥v∥ = 1, (v, ϕj ) = 0, j = 1, . . . , n − 1
}
.
(6.13)
Note that the process does not stop after a finite number of steps. For, if
(v, ϕj ) = 0, j = 1, . . . , n − 1, were to imply v = 0, then L2 would be finite-
dimensional, which it is not. One may show the following.
Theorem 6.3. With λn the n
th eigenvalue of (6.5) we have that λn → ∞
as n → ∞.
The proof of this theorem will also be postponed till later.
As a result of this theorem a number in the nondecreasing sequence
{λj}
∞
j=1 can only be repeated a finite number of times. If λn−1 < λn = λn+1 = · · · = λn+m−1 < λn+m, then we say that the eigenvalue λn has mul- tiplicity m. The set En of linear combinations of ϕn, . . . , ϕn+m−1 is then a finite-dimensional linear space of dimension m, the eigenspace corresponding to λn. For v ∈ En we thus have −∆v = λnv. We remark that the first, or principal, eigenvalue λ1 is a simple eigenvalue, i.e., that λ2 > λ1, and that the corresponding principal eigenfunction ϕ1 may
be chosen to be positive in Ω, after a possible change of sign. To indicate a
proof of this, we write ϕ1 = ϕ = ϕ
+ − ϕ−, where ϕ± = max(±ϕ, 0). One
may show that ϕ± ∈ H10 and that ∇ϕ
+ = ∇ϕ when ϕ ≥ 0 and ∇ϕ+ = 0
when ϕ < 0 so that a(ϕ+, ϕ−) = (∇ϕ+, ∇ϕ−) = 0. We then have that ∥∇ϕ±∥2 = λ1∥ϕ ±∥2, since otherwise λ1 = ∥∇ϕ∥ 2 = ∥∇ϕ+∥2 + ∥∇ϕ−∥2 > λ1
(
∥ϕ+∥2 + ∥ϕ−∥2
)
= λ1∥ϕ∥
2 = λ1,
which is a contradiction. Hence ϕ± both satisfy −∆ϕ± = λ1ϕ
±. But then
−∆ϕ+ = λ1ϕ
+ ≥ 0, and hence the strong maximum principle shows that
ϕ+ > 0 in Ω or ϕ+ = 0 in Ω, so that ϕ1 > 0 in Ω or ϕ1 < 0 in Ω. This also implies that λ1 is a simple eigenvalue, since there cannot exist two orthogonal eigenfunctions with constant sign. We now turn to the question of how eigenfunctions may be used for series expansions of other functions and consider the case of a general Hilbert space H. Let {ϕj} ∞ j=1 be an orthonormal sequence, i.e., one for which (ϕi, ϕj ) = δij = { 1, if i = j, 0, if i ̸= j. Such a sequence is called an orthonormal basis (or a complete orthonormal set) if any v in H can be approximated arbitrarily well by a linear combination of elements from the sequence, i.e., if for any ϵ > 0 there exist an integer N
and real numbers α1, . . . , αN such that

82 6 The Elliptic Eigenvalue Problem
∥
∥
∥v −
N∑
j=1
αj ϕj
∥
∥
∥ < ϵ. Note that it suffices to show this for v in a dense subset M of H. In fact, let v ∈ H. That M is a dense subset of H means that one may find w ∈ M such that ∥v − w∥ < ϵ/2. Therefore it suffices to find a linear combination such that ∥ ∥ ∥w − N∑ j=1 αj ϕj ∥ ∥ ∥ < ϵ/2, since then ∥ ∥ ∥v − N∑ j=1 αj ϕj ∥ ∥ ∥ ≤ ∥v − w∥ + ∥ ∥ ∥w − N∑ j=1 αj ϕj ∥ ∥ ∥ < ϵ. Lemma 6.1. Let {ϕj} ∞ j=1 be an orthonormal set in H. Then the best ap- proximation of v ∈ H by a linear combination of the first N functions ϕj is vN = ∑N j=1(v, ϕj )ϕj . Proof. We have, for arbitrary α1, . . . , αN , ∥ ∥ ∥v − N∑ j=1 αj ϕj ∥ ∥ ∥ 2 = ∥v∥2 − 2 N∑ j=1 αj (v, ϕj ) + N∑ j=1 a2j = ∥v∥2 + N∑ j=1 ( αj − (v, ϕj ) )2 − N∑ j=1 (v, ϕj ) 2, from which the result follows at once. ⊓. Since the left-hand side is nonnegative, we obtain, in particular, for αj = (v, ϕj ), N∑ j=1 (v, ϕj ) 2 ≤ ∥v∥2. Since this holds for each N we infer Bessel’s inequality ∞∑ j=1 (v, ϕj ) 2 ≤ ∥v∥2. If {ϕj} ∞ j=1 is an orthonormal basis, then the error in the best approxima- tion has to tend to zero as N tends to infinity, so that (6.14) ∥ ∥ ∥v − N∑ j=1 (v, ϕj )ϕj ∥ ∥ ∥ 2 = ∥v∥2 − N∑ j=1 (v, ϕj ) 2 → 0, as N → ∞. 6.1 Eigenfunction Expansions 83 This is equivalent to Parseval’s relation ∞∑ j=1 (v, ϕj ) 2 = ∥v∥2. Thus the orthonormal set {ϕj} ∞ j=1 is an orthonormal basis of H if and only if Parseval’s relation holds for all v in H (or for all v in a dense subset of H). We return to the concrete case of H = L2. We then have the following. Theorem 6.4. The eigenfunctions {ϕj} ∞ j=1 of (6.5) form an orthonormal basis for L2. The series ∑∞ j=1 λj (v, ϕj ) 2 is convergent if and only if v ∈ H10 . Moreover, (6.15) ∥∇v∥2 = a(v, v) = ∞∑ j=1 λj (v, ϕj ) 2, for all v ∈ H10 . Proof. By our above discussion it follows that for the first statement it suffices to show (6.14) for all v in H10 , which is a dense subspace of L2. We shall demonstrate that (6.16) ∥ ∥ ∥v − N∑ j=1 (v, ϕj )ϕj ∥ ∥ ∥ ≤ Cλ−1/2N+1 , for all v ∈ H 1 0 , which then implies (6.14) in view of Theorem 6.3. To prove (6.16), set vN = ∑N j=1(v, ϕj )ϕj and rN = v − vN . Then (rN , ϕj ) = 0 for j = 1, . . . , N , so that ∥∇rN ∥ 2 ∥rN ∥2 ≥ inf { ∥∇v∥2 : v ∈ H10 , ∥v∥ = 1, (v, ϕj ) = 0, j = 1, . . . , N } = λN+1, and hence ∥rN ∥ ≤ λ −1/2 N+1 ∥∇rN ∥. It now suffices to show that the sequence ∥∇rN ∥ is bounded. We first recall from Theorem 6.1 that a(ϕi, ϕj ) = 0 for i ̸= j, so that a(rN , vN ) = 0. Hence a(v, v) = a(vN , vN ) + 2a(vN , rN ) + a(rN , rN ) = a(vN , vN ) + a(rN , rN ) and ∥∇rN ∥ 2 = a(rN , rN ) = a(v, v) − a(vN , vN ) ≤ a(v, v) = ∥∇v∥ 2, which completes the proof of (6.16). For the proof of the second statement, we first note that, for v ∈ H10 , N∑ j=1 λj (v, ϕj ) 2 = a(vN , vN ) = a(v, v) − a(rN , rN ) ≤ a(v, v), and we conclude that ∑∞ j=1 λj (v, ϕj ) 2 < ∞. Conversely, we assume that v ∈ L2 and ∑∞ j=1 λj (v, ϕj ) 2 < ∞. We already know that vN → v in L2 as 84 6 The Elliptic Eigenvalue Problem N → ∞. To obtain convergence in H1 we note that, with M > N , in view
of (6.8),
α∥vN − vM ∥
2
1 ≤ ∥∇(vN − vM )∥
2 =
M∑
j=N+1
λj (v, ϕj )
2
→ 0 as N → ∞.
Hence, vN is a Cauchy sequence in H
1 and converges to a limit in H1. Clearly,
this limit is the same as v. By the trace theorem (Theorem A.4) vN is also a
Cauchy sequence in L2(Γ ), and since vN = 0 on Γ we conclude that v = 0
on Γ . Hence, v ∈ H10 . Finally, (6.15) is obtained by letting N → ∞ in
a(vN , vN ) =
∑N
j=1 λj (v, ϕj )
2. ⊓.
It follows immediately from (6.13) that
(6.17) λn = min
(v,ϕj )=0,
j=1,…,n−1
a(v, v)
∥v∥2
.
This in turn implies the following min-max principle.
Theorem 6.5. We have
(6.18) λn = min
Vn
max
v∈Vn
a(v, v)
∥v∥2
,
where Vn varies over all subspaces of H
1
0 of finite dimension n.
Proof. Let En denote the n-dimensional subspace of linear combinations v =∑n
j=1 αj ϕj of the eigenfunctions ϕ1, . . . , ϕn. Then clearly
max
v∈En
a(v, v)
∥v∥2
= max
α1,…,αn
∑n
j=1 α
2
j λj∑n
j=1 α
2
j
= λn,
where the maximum is attained by ϕn. It therefore remains to show that for
any Vn of dimension n,
max
v∈Vn
a(v, v)
∥v∥2
≥ λn.
To see this we choose w ∈ Vn such that
(w, ϕj ) = 0, for j = 1, . . . , n − 1.
If {ψj}
n
j=1 is a basis for Vn, then such a w =
∑n
j=1 αj ψj may be determined
from the linear system of equations
(w, ϕj ) =
n∑
l=1
αl(ψl, ϕj ) = 0, j = 1, . . . , n − 1,

6.1 Eigenfunction Expansions 85
which has a nonzero solution since the number of equations is smaller than
n. By (6.17) it follows that
a(w, w)
∥w∥2
≥ λn,
which thus completes the proof of (6.18). ⊓.
One consequence of this result is that the eigenvalues depend monotonous-
ly on the underlying domain. More precisely, if Ω ⊂ Ω̃ and the corresponding
eigenvalues are λn(Ω) and λn(Ω̃), then we have λn(Ω̃) ≤ λn(Ω) for all n.
In fact, by extending the functions in H10 (Ω) by zero in Ω̃ \ Ω, we have
H10 (Ω) ⊂ H
1
0 (Ω̃), and hence the minimum in the expression for λn(Ω) in
(6.18) is taken over a smaller set of n-dimensional spaces than for λn(Ω̃),
and hence the latter minimum is at least as small.
We now return to the more subtle mathematical points that we postponed
earlier. For their treatment we shall need to use the concept of compactness,
which we first briefly discuss.
We say that a set M in a Hilbert space H (with norm ∥·∥) is pre-compact,
if every infinite sequence {un}
∞
n=1 ⊂ M contains a convergent subsequence,
i.e., there is a subsequence {unj }
∞
j=1 and an element ū ∈ H such that
(6.19) ∥unj − ū∥ → 0, as j → ∞.
As an example we recall from elementary calculus that a bounded infinite
sequence of real numbers is pre-compact (the Bolzano-Weierstrass theorem).
The set M is called compact, if it is also a closed set, i.e., if the limit ū
in (6.19) always belongs to M. Below we shall need the following result for
H10 = H
1
0 (Ω) with Ω ⊂ R
d, the proof of which is beyond the scope of this
book.
Lemma 6.2. (Rellich’s lemma.) A bounded subset M of H1 is pre-compact
in L2.
Thus if {un}
∞
n=1 ⊂ H
1 and ∥un∥1 ≤ C for n ≥ 1, then there exists a
subsequence {unj }
∞
j=1 and ū ∈ L2 such that (6.19) holds in L2-norm.
We shall now use this to prove the first statement of Theorem 6.2, that
the infimum in (6.10) is attained in H10 . For this, we take a sequence {un}
∞
n=1
such that
(6.20) ∥∇un∥
2 = a(un, un) → λ1 and ∥un∥ = 1, as n → ∞,
which is possible by the definition of the infimum. Then clearly {un}
∞
n=1 is
bounded in H1, and, by Lemma 6.2, we may therefore take a subsequence,
which converges to an element ϕ1 ∈ L2. By changing the notation if necessary,
we may assume that {un}
∞
n=1 itself is this subsequence, so that ∥un−ϕ1∥ → 0.

86 6 The Elliptic Eigenvalue Problem
We now want to show that {un}
∞
n=1 converges in H
1
0 . By a simple calcu-
lation we have
∥∇(un − um)∥
2 = 2∥∇un∥
2 + 2∥∇um∥
2
− 4∥ 1
2
∇(un + um)∥
2
(the parallellogram law), and by the definition of λ1,
∥
1
2
∇(un + um)∥
2
≥ λ1∥
1
2
(un + um)∥
2.
Hence
(6.21) ∥∇(un − um)∥
2
≤ 2∥∇un∥
2 + 2∥∇um∥
2
− 4λ1∥
1
2
(un + um)∥
2.
It is clear that ∥ 1
2
(un + um)∥ → ∥ϕ1∥ as n, m → ∞, and since ∥un∥ = 1,
we have ∥ϕ1∥ = 1. Thus, by (6.20) the right-hand side of (6.21) tends to
zero, so that {un}
∞
n=1 is a Cauchy sequence in H
1
0 , i.e., ∥∇(un − um)∥ → 0
as m, n → ∞. Since H10 is a Hilbert space, the sequence thus converges to an
element of H10 , which then has to be the same as the limit in L2, i.e., ϕ1. In
particular,
∥∇ϕ1∥
2 = λ1,
which shows that ϕ1 realizes the minimum in (6.10).
The proof that the infimum is attained in (6.12) is analogous. In fact,
if {un}
∞
n=1 is a minimizing sequence that converges to some ϕ2 in L2 and
satisfies the side conditions in (6.12), then, since ( 1
2
(un + um), ϕ1) = 0, we
have now (6.21) with λ1 replaced by λ2, and we conclude that un converges
in H10 to ϕ2, and that
∥ϕ2∥ = 1, (ϕ2, ϕ1) = 0, ∥∇ϕ2∥
2 = λ2.
We finally give the proof of Theorem 6.3. Assume then that the result is
not valid, so that
∥∇ϕn∥
2 = λn ≤ C, for n ≥ 1.
But then, by compactness, {ϕn}
∞
n=1 contains a subsequence {ϕnj }
∞
j=1, which
converges in L2. But since {ϕn}
∞
n=1 is orthonormal we have
∥ϕi − ϕj∥
2 = ∥ϕ1∥
2 + ∥ϕ2∥
2 = 2, for i ̸= j,
so that no convergent subsequence can exist.
As we mentioned before, the theory for the more general eigenvalue prob-
lem (6.9) is analogous. For example, for the eigenvalue problem related
to the Neumann problem in Sect. 3.6, our theorems hold with a(v, w) =∫
Ω
(
a ∇v · ∇w + c vw
)
dx and H10 replaced by V = H
1. In particular,
λ1 = min
v∈V
∫
Ω
(
a |∇v|2 + c v2
)
dx
∫
Ω
v2 dx
.

6.1 Eigenfunction Expansions 87
We close by two examples where we can solve the eigenvalue problem
explicitly.
Example 6.1. Let Ω = (0, b) ⊂ R. The problem (6.5) then reduces to
(6.22) −u′′ = λu in Ω, with u(0) = u(b) = 0.
Here we may easily determine the eigenfunctions and eigenvalues explicitly.
In fact, the general solution of the differential equation in (6.22) is
u = C1 sin(
√
λx) + C2 cos(
√
λx), with λ > 0,
and the boundary conditions show that C2 = 0 and
√
λb = nπ. Hence the
eigenfunctions are {sin(nπx/b)}∞n=1 and the corresponding eigenvalues λn =
n2π2/b2. After normalization we thus find ϕn(x) =
√
2/b sin(nπx/b), n =
1, 2, . . . , for our orthonormal basis of eigenfunctions in L2(Ω). Note in par-
ticular that the eigenvalues decrease with increasing b.
Example 6.2. Let Ω = (0, b) × (0, b) and consider the eigenvalue problem
−∆u = λu in Ω, with u = 0 on Γ.
Then with λn and ϕn as in Example 6.1 it is easy to check that the products
φml(x) = ϕm(x1)ϕl(x2), m, l = 1, 2, . . . , are eigenfunctions corresponding to
the eigenvalues λml = (m
2 + l2)π2/b2. To see that these are all the eigenfunc-
tions, it suffices to show Parseval’s relation
∑∞
m,l=1(φml, v)
2 = ∥v∥2, i.e.,
(6.23)
∞∑
m,l=1
(∫
Ω
v(x)ϕm(x1)ϕl(x2) dx
)2
=
∫
Ω
v(x)2 dx.
But, using Parseval’s relation in x2 we have
(6.24)
∫ b
0
v(x1, x2)
2 dx2 =
∞∑
l=1
wl(x1)
2, wl(x1) =
∫ b
0
v(x1, x2)ϕl(x2) dx2.
Applying Parseval’s relation to wl(x1) we find
∫ b
0
wl(x1)
2 dx1 =
∞∑
m=1
(wl, ϕm)
2
=
∞∑
m=1
(∫ b
0
∫ b
0
v(x1, x2)ϕm(x1)ϕl(x2) dx1 dx2
)2
.
(6.25)
We now integrate (6.24) in x1 and insert (6.25) to obtain (6.23).
This shows that the eigenvalues are the numbers λml = (m
2 + l2)π2/b2,
arranged in increasing order, and with multiple eigenvalues repeated. To de-
termine the rate of growth of the eigenvalues λn as n increases, we observe

88 6 The Elliptic Eigenvalue Problem
that the number of eigenvalues with λn ≤ ρ
2 is equal to the number of mesh-
points (mπ/b, lπ/b) in the disc Dρ = {x
2
1 + x
2
2 ≤ ρ
2}. Since the number Nρ
of such mesh-points equals the number of mesh-squares with area π2/b2 that
can be fitted into Dρ, we have Nρ ≈ ρ
2b2/π. Hence, for λn corresponding to
λml, we have λn = λml ≈ ρ
2 ≈ πNρ/b
2 ≈ πn/b2.
Since any domain Ω ⊂ R2 contains a square and is contained in another
square, it follows by the monotonicity of the eigenvalues that for any domain
Ω, there are positive constants c and C such that cn ≤ λn ≤ Cn. In d
dimensions the corresponding inequality is cn2/d ≤ λn ≤ Cn
2/d.
6.2 Numerical Solution of the Eigenvalue Problem
We shall first consider the one-dimensional eigenvalue problem (6.1), i.e.,
(6.26) Aϕ := −(aϕ′)′ + cϕ = λϕ in Ω = (0, 1), with ϕ(0) = ϕ(1) = 0,
where a and c are smooth functions such that a(x) ≥ a0 and c(x) ≥ 0 on Ω̄.
To formulate a finite difference discretization we use the notation of Sect. 4.1
based on the mesh-points xj = jh, j = 0, . . . , M , where h = 1/M , with
Uj ≈ u(xj ), and consider the finite-dimensional eigenvalue problem
(6.27)
AhUj := − ∂̄(aj+1/2∂Uj ) + cj Uj = ΛUj , j = 1, . . . , M − 1,
U0 = UM = 0,
where cj = c(xj ) and aj+1/2 = a(xj + h/2). The equation at the interior
mesh-point xj may then be written
−
(
aj+1/2Uj+1 − (aj+1/2 + aj−1/2)Uj + aj−1/2Uj−1
)
/h2 + cj Uj = ΛUj ,
and thus, with A an (M − 1) × (M − 1) tridiagonal matrix and Ū =
(U1, . . . , UM−1) ∈ R
M−1 the vector corresponding to the interior mesh-
points, (6.27) may be expressed as the matrix eigenvalue problem
A Ū = Λ Ū .
For the analysis we introduce a discrete inner product and a norm by
(V, W )h = h
M∑
j=0
Vj Wj and ∥V ∥h = (V, V )
1/2
h ,
respectively. The operator Ah is easily seen to be symmetric with respect to
this inner product, and positive definite since
(AhU, U )h = h
M−1∑
j=1
AhUj Uj = h
M−1∑
j=0
aj+1/2(∂Uj )
2 + h
M−1∑
j=1
cj U
2
j .

6.2 Numerical Solution of the Eigenvalue Problem 89
It would be natural to expect the eigenvalues of the discrete problem
(6.27) (or of the matrix A) to approximate those of the continuous eigenvalue
problem (6.26). We shall show this for the principal eigenvalue Λ1 only.
Theorem 6.6. Let Λ1 and λ1 be the smallest eigenvalues of (6.27) and
(6.26). Then
|Λ1 − λ1| ≤ Ch
2.
Proof. We carry out the proof for c = 0 only and leave the general case
to Problem 6.4. Let U ∈ RM+1 be arbitrary with U0 = UM = 0, and let
ũ = IhU ∈ C(Ω̄) be the associated piecewise linear interpolant. Then
(6.28) λ1 ≤
a(ũ, ũ)
∥ũ∥2
.
Since ũ′ = ∂Uj in (xj , xj+1) we have, using a ≥ a0 > 0 in Ω, that
a(ũ, ũ) =
M−1∑
j=0
∫ xj+1
xj
a dx (∂Uj )
2
≤ h
M−1∑
j=0
(aj+1/2 + Ch
2)(∂Uj )
2
≤ (AhU, U )h(1 + Ch
2).
Further, by simple calculations,
∥ũ∥2 =
M−1∑
j=0
h−2
∫ xj+1
xj
(
(xj+1 − x)Uj + (x − xj )Uj+1
)2
dx
= 1
3
h
M−1∑
j=0
(U 2j + Uj Uj+1 + U
2
j+1)
and, since U0 = UM = 0,
(6.29) ∥U∥2h =
1
2
h
M−1∑
j=0
(U 2j + U
2
j+1).
Hence, since aj+1/2 ≥ a0, cj ≥ 0,
∥U∥2h − ∥ũ∥
2 = 1
6
h
M−1∑
j=0
(Uj − Uj+1)
2 = 1
6
h3
M−1∑
j=0
(∂Uj )
2
≤ Ch2(AhU, U )h,
or
∥U∥2h ≤ ∥ũ∥
2 + Ch2(AhU, U )h.
Hence, if U is chosen as an eigenvector corresponding to Λ1, with ∥U∥h = 1,
then we have for small h,
a(ũ, ũ)
∥ũ∥2
≤
(AhU, U )h(1 + Ch
2)
∥U∥2h − Ch
2(AhU, U )h
=
Λ1(1 + Ch
2)
1 − CΛ1h2
≤ Λ1 + Ch
2,

90 6 The Elliptic Eigenvalue Problem
so that, by (6.28),
(6.30) λ1 ≤ Λ1 + Ch
2.
To show the converse inequality, note that for u smooth and U defined as
the restriction of u to the mesh-points, we have
(AhU, U )h = a(u, u) + O(h
2) and ∥U∥2h = ∥u∥
2 + O(h2), as h → 0.
For the second relation we have used that, since U0 = UM = 0, (6.29) is
the second order trapezoidal quadrature rule for
∫ 1
0
u2 dx. In particular, with
u = ϕ1, the principal eigenfunction of the continuous problem, we have
Λ1 ≤
(AhU, U )h
∥U∥2h
≤
a(ϕ1, ϕ1) + Ch
2
∥ϕ1∥ − Ch2
≤ λ1 + Ch
2, for h small,
Together with (6.30) this completes the proof. ⊓.
Consider now the eigenvalue problem
(6.31) −∆u = λu in Ω, with u = 0 on Γ,
where Ω ⊂ R2, and let λn be the n
th eigenvalue and ϕn the corresponding
eigenfunction.
For the case that Ω is the square (0, 1) × (0, 1) we may use the five-point
approximation −∆h defined in Chapt. 4 and pose the discrete eigenvalue
problem
−∆hu = Λu in Ω, with u = 0 on Γ.
This may be treated as our above one-dimensional problem, but this is not so
interesting, since the eigenvalues of (6.31) can be determined directly, as we
have seen in Example 6.2. When Ω is a more general domain with a curved
smooth boundary Γ , we encounter, as for the Dirichlet problem discussed
in Chapt. 4, the difficulties caused by the fact that the uniform mesh does
not fit the domain. The analysis therefore becomes involved and we shall not
pursue it here.
In this case, the finite element method, with its greater flexibility, is better
suited, and we shall give an elementary presentation of some simple results.
We assume thus for simplicity that Ω ⊂ R2 is a convex domain with smooth
boundary Γ and denote by {Sh} a family of spaces of continuous piecewise
linear functions based on regular triangulations Th. The corresponding dis-
crete eigenvalue problem is then
(6.32) a(uh, χ) = λ(uh, χ), ∀χ ∈ Sh, where a(v, w) = (∇v, ∇w).
Using the basis {Φi}
Mh
i=1 of pyramid functions from Sect. 5.2, and the positive
definite matrices A and B with elements aij = (∇Φi, ∇Φj ) and bij = (Φi, Φj ),
respectively, this problem may be written in matrix form as

6.2 Numerical Solution of the Eigenvalue Problem 91
(6.33) A U = λB U.
Note that in contrast to the finite difference case, the matrix B is not diagonal.
Nevertheless, the eigenvalue problem (6.32) or (6.33) has positive eigenvalues
{λn,h}
Mh
n=1 and orthonormal eigenfunctions {ϕn,h}
Mh
n=1. In this case we have
first the following error estimates for the eigenvalues.
Theorem 6.7. Let λn,h and λn be the n
th eigenvalues of (6.32) and (6.31),
respectively. Then there are constants C and h0 (depending on n) such that
(6.34) λn ≤ λn,h ≤ λn + Ch
2, for h ≤ h0.
Proof. By the min-max principle in Theorem 6.5
λn = min
Vn⊂H10
max
v∈Vn
∥∇v∥2
∥v∥2
, dim Vn = n.
Similarly,
(6.35) λn,h = min
Vn⊂Sh
max
χ∈Vn
∥∇χ∥2
∥χ∥2
, dim Vn = n.
Since Sh ⊂ H
1
0 , the minimum in the latter expression is taken over a smaller
set of subspaces than the former, and hence is at least as large, which shows
the first inequality in the theorem.
To show the second inequality we note that, with En the space spanned
by ϕ1, . . . , ϕn and En,h = RhEn, where Rh is the Ritz projection,
(6.36) λn,h ≤ max
χ∈En,h
∥∇χ∥2
∥χ∥2
= max
v∈En
∥∇Rhv∥
2
∥Rhv∥2
≤ max
v∈En
∥∇v∥2
∥Rhv∥2
,
since ∥∇Rhv∥ ≤ ∥∇v∥. To estimate the denominator, we have
∥Rhv∥ ≥ ∥v∥ − ∥Rhv − v∥.
Here, for v ∈ En, using Theorem 5.5 and the regularity estimate (3.36),
∥Rhv − v∥ ≤ Ch
2
∥v∥2 ≤ Ch
2
∥∆v∥ ≤ Ch2λn∥v∥ ≤ Ch
2
∥v∥,
where we have used that n is fixed. Hence
∥Rhv∥ ≥ ∥v∥(1 − Ch
2),
and it follows from (6.36), for h small,
λn,h ≤ max
v∈En
∥∇v∥2
∥v∥2
(1 + Ch2) ≤ λn + Ch
2,
which completes the proof. ⊓.

92 6 The Elliptic Eigenvalue Problem
A property that is sometimes used for finite element spaces {Sh} is the
so-called inverse inequality
(6.37) ∥∇χ∥ ≤ Ch−1∥χ∥ for χ ∈ Sh.
In particular, this is valid for piecewise linear finite element spaces based on
a quasi-uniform family of triangulations {Th}, see Problem 6.6. When this
holds it follows immediately from (6.35) that the largest eigenvalue satisfies
(6.38) λMh,h = max
χ∈Sh
∥∇χ∥2
∥χ∥2
≤ Ch−2.
One may also derive error estimates for the eigenfunctions. We do this
only for the first eigenvalue, because we want to avoid the complications that
arise for multiple eigenvalues.
Theorem 6.8. Let ϕ1,h and ϕ1 be normalized eigenfunctions corresponding
to the principal eigenvalues of (6.32) and (6.31), respectively. Then
(6.39) ∥ϕ1,h − ϕ1∥ ≤ Ch
2
and
(6.40) ∥∇ϕ1,h − ∇ϕ1∥ ≤ Ch.
Proof. We expand Rhϕ1 in discrete eigenfunctions,
Rhϕ1 =
Mh∑
j=1
aj ϕj,h, where aj = (Rhϕ1, ϕj,h),
and conclude by Parseval’s relation
(6.41) ∥Rhϕ1 − a1ϕ1,h∥
2 =
Mh∑
j=2
a2j .
Using (6.32) we find
λj,haj = λj,h(Rhϕ1, ϕj,h) = a(Rhϕ1, ϕj,h) = a(ϕ1, ϕj,h) = λ1(ϕ1, ϕj,h),
and hence
(λj,h − λ1)aj = λ1(ϕ1 − Rhϕ1, ϕj,h).
Using the first inequality in (6.34) and the fact that λ1 is a simple eigenvalue,
we have λjh − λ1 ≥ λ2 − λ1 > 0 for j ≥ 2, and we may conclude
Mh∑
j=2
a2j ≤
Mh∑
j=2
( λ1
λj,h − λ1
)2
(ϕ1 − Rhϕ1, ϕj,h)
2
≤ C∥Rhϕ1 − ϕ1∥
2
≤ Ch4,

6.3 Problems 93
so that by (6.41)
∥Rhϕ1 − a1ϕ1,h∥ ≤ Ch
2.
We therefore have,
(6.42) ∥a1ϕ1,h − ϕ1∥ ≤ ∥Rhϕ1 − ϕ1∥ + ∥Rhϕ1 − a1ϕ1,h∥ ≤ Ch
2,
and it thus remains to bound ∥a1ϕ1,h −ϕ1,h∥ = |a1 −1|. We may assume that
the sign of ϕ1,h is chosen so that a1 ≥ 0. Then, by the triangle inequality
and (6.42),
|a1 − 1| =
∣
∣∥a1ϕ1,h∥ − ∥ϕ1∥
∣
∣ ≤ ∥a1ϕ1,h − ϕ1∥ ≤ Ch2,
which completes the proof of (6.39).
We now turn to the error in the gradient. We have, using (6.32) and the
error bounds already derived,
∥∇ϕ1,h − ∇ϕ1∥
2 = ∥∇ϕ1,h∥
2
− 2(∇ϕ1,h, ∇ϕ1) + ∥∇ϕ1∥
2
= λ1,h − 2λ1(ϕ1,h, ϕ1) + λ1 = λ1,h − λ1 + λ1∥ϕ1,h − ϕ1∥
2
≤ Ch2,
which shows (6.40) and thus completes the proof of the theorem. ⊓.
6.3 Problems
Problem 6.1. Consider the problem (6.1).
(a) Show that if the functions a(x) and c(x) are increased then all the corre-
sponding eigenvalues increase.
(b) Find the eigenvalues when a(x) and c(x) are constant on Ω.
(c) Show that for given a(x) and c(x) there are constants k1 and k2 such
that
0 < k1n 2 ≤ λn ≤ k2n 2. Problem 6.2. Consider the Laplacian in spherical symmetry, see Prob- lem 1.4. Then the corresponding eigenvalue problem is − 1 r2 d dr ( r2 dϕ dr ) = λϕ for 0 < r < 1, with ϕ(1) = 0, ϕ(0) finite. Prove that the eigenfunctions ϕj of (6.2), corresponding to different eigen- values λi and λj , satisfy ∫ 1 0 ϕi(r)ϕj (r) r 2 dr = ∫ 1 0 ϕ′i(r)ϕ ′ j (r) r 2 dr = 0, i.e., {ϕi} ∞ i=1 is an orthogonal set in L2((0, 1); r 2 dr), the set of functions that are square integrable on (0, 1) with respect to the measure r2 dr. Prove also that, properly normalized, the functions ϕi form an orthonormal basis for L2((0, 1); r 2 dr). 94 6 The Elliptic Eigenvalue Problem Problem 6.3. Assume that Ω is such that (3.36) holds. (a) Use an argument similar to that of Theorem 6.4 to show that v ∈ H2 ∩ H10 if and only if ∞∑ i=1 λ2i (v, ϕi) 2 < ∞. (b) Show that −∆v = ∞∑ i=1 λi(v, ϕi)ϕi, ∥∆v∥ 2 = ∞∑ i=1 λ2i (v, ϕi) 2, for v ∈ H2 ∩ H10 . Problem 6.4. Prove Theorem 6.6 in the general case when the function c(x) ≥ 0 does not necessarily vanish. Problem 6.5. Show that the largest eigenvalue of (6.27) satisfies ΛM−1 ≤ CM 2, with C independent of M . Problem 6.6. Show the inverse inequality (6.37) for piecewise linear finite element functions based on a family {Th} of quasi-uniform triangulations of a plane domain, see (5.52). Hint: Make an affine transformation x = Ax̂ + b from the small triangle K to a fixed reference triangle K̂ of unit size, see Problem A.15, and use the fact that the norms ∥ · ∥L2(K̂) and ∥ · ∥H1(K̂) are equivalent on the finite-dimensional space Π1. Problem 6.7. Let G the Green’s function in (3.18) of Sect. 3.4 and let {λj} ∞ j=1 and {ϕj} ∞ j=1 be the eigenvalues and normalized eigenfunctions of (6.5) as in Theorem 6.4. Show that G(x, y) = ∞∑ j=1 λ−1j ϕj (x)ϕj (y). Problem 6.8. Discuss the eigenvalue problem related to the Neumann prob- lem in Problem 3.9. Hint: The smallest eigenvalue is λ1 = 0. 7 Initial-Value Problems for Ordinary Differential Equations As a preparation for our study of initial-value problems for parabolic and hyperbolic differential equations we shall review in this chapter some facts about linear systems of ordinary differential equations and their numerical solution. We start with the continuous problem in Sect. 7.1 and continue in Sect. 7.2 with the numerical solution of such problems by time stepping. 7.1 The Initial Value Problem for a Linear System We first consider the initial-value problem for the first order scalar linear ordinary differential equation (7.1) u′ + au = f (t), for t > 0, with u(0) = v,
where a is a constant, f (t) a given smooth function, and v a given num-
ber. We recall from elementary calculus that this problem may be solved by
multiplication by the integrating factor eat, which gives
(eatu)′ = eatf (t),
from which
eatu(t) = v +
∫ t
0
easf (s) ds,
or
(7.2) u(t) = e−atv +
∫ t
0
e−a(t−s)f (s) ds.
We consider now the corresponding problem for a system of equations
u′i +
N∑
j=1
aij uj = fi(t), i = 1, . . . , N, for t > 0,
ui(0) = vi, i = 1, . . . , N.
Introducing the column vector u = (u1, . . . , uN )
T, and similarly for f (t) and
v, and also the matrix A = (aij ), this may be written

96 7 Initial-Value Problems for ODEs
(7.3) u′ + Au = f (t), for t > 0, with u(0) = v.
We now want to generalize the above solution method in the scalar case
to the system (7.3). For this purpose we first define the exponential of an
N × N matrix B = (bij ) by means of the power series
eB = exp(B) =
∞∑
j=0
1
j!
Bj ,
where B0 = I, the identity matrix. This definition is based on the Maclaurin
expansion of ex, and it is easily shown that the series converges for any matrix
B. We note that if B1 and B2 are two N × N matrices which commute, i.e.,
such that B1B2 = B2B1, then
(7.4) eB1+B2 = eB1 eB2 = eB2 eB1
In fact, since B1 and B2 commute, we have
(B1 + B2)
j =
j∑
l=0
(
j
l
)
Bl1B
j−l
2 ,
and hence, formally,
eB1+B2 =
∞∑
j=0
1
j!
j∑
l=0
(
j
l
)
Bl1B
j−l
2 =
∞∑
j=0
j∑
l=0
1
l!(j − l)!
Bl1B
j−l
2
=
∞∑
l=0
∞∑
m=0
1
l! m!
Bl1B
m
2 =
∞∑
l=0
1
l!
Bl1
∞∑
m=0
1
m!
Bm2 = e
B1 eB2 .
Note that if B1 and B2 do not commute, then we have, for instance,
(B1 + B2)
2 = B21 + B1B2 + B2B1 + B
2
2 ̸= B
2
1 + 2B1B2 + B
2
2 .
Considering the matrix e−tA we have for its derivative
d
dt
e−At =
d
dt
∞∑
j=0
1
j!
tj (−A)j =
∞∑
j=1
1
(j − 1)!
tj−1(−A)j = −Ae−tA,
and hence u(t) = e−tAv satisfies
(7.5) u′ + Au = 0, for t > 0, with u(0) = v.
Multiplication by e−tA may thus be thought of as an operator E(t), the
solution operator of (7.5), that takes the initial data v of this problem into
the solution at time t, so that u(t) = E(t)v = e−tAv. Note that, by (7.4),

7.1 The Initial Value Problem for a Linear System 97
E(t + s) = e−(t+s)A = e−tAe−sA = E(t)E(s), for s, t ≥ 0
which is referred to as the semigroup property of E(t). This expresses the
fact that the solution of (7.5) at time t + s may be obtained by using the
solution at time s as the initial value for (7.5) and then looking at its solution
at time t. Note also that it follows from the above that A commutes with
E(t), so that AE(t) = E(t)A.
To solve the inhomogeneous equation (7.3) analogously to the above we
multiply the equation by etA to obtain
etA(u′ + Au) = etAf (t), for t > 0.
This may be written
d
dt
(etAu) = etAf (t),
and hence by integration
etAu(t) = v +
∫ t
0
esAf (s) ds.
Multiplying by e−tA and using (7.4), we obtain the formula analogous to
(7.2),
(7.6) u(t) = e−tAv +
∫ t
0
e−(t−s)Af (s) ds.
In terms of the solution operator introduced above this may also be expressed
as
(7.7) u(t) = E(t)v +
∫ t
0
E(t − s)f (s) ds.
We note that the integrand E(t−s)f (s) is the solution at t−s of the homoge-
neous equation in (7.5) with initial data f (s). The integral may therefore be
interpreted as the superposition of the solutions of these initial value prob-
lems, and (7.7) is often referred to as Duhamel’s principle.
In some cases the system (7.3) may be reduced to a finite number of
independent scalar equations of type (7.1). To see this, we assume that A is
such that there is a diagonal matrix Λ and a nonsingular matrix P such that
A = P ΛP −1. We may then introduce the new dependent variable w = P −1u
and the source term g = P −1f to find, after multiplication by P −1, that the
equation (7.3) may be written
w′ + Λw = g(t), for t > 0, with w(0) = P −1v.
Denoting the diagonal elements of Λ by λi we may write this as

98 7 Initial-Value Problems for ODEs
w′i + λiwi = gi(t), i = 1, . . . , N, for t > 0,
These equations may now be solved individually and we find the solution of
our original problem by taking u = P w.
The assumption that A may be transformed as above to a diagonal matrix
is satisfied, for example, if A is symmetric (or selfadjoint), i.e., if aij = aji
for all i, j, in which case A = P ΛP T, where P is an orthogonal matrix, so
that P T = P −1. In any case, the elements of Λ are the eigenvalues of A,
and the method applies when A has N linearly independent eigenvectors.
For large N this is not necessarily a good method for practical calculations
as the diagonalization of A could be costly.
We shall now briefly study how the solutions behave for large t, and
restrict ourselves to the case that A is symmetric. Let thus A = P ΛP T,
where P is an orthogonal matrix and Λ is the diagonal matrix whose diagonal
entries are the eigenvalues λj of A, which are real. Recall that the j
th column
of P is the eigenvector corresponding to λj . Then, since P
TP = P P T = I,
e−tA =
∞∑
j=0
1
j!
(−P ΛP T)j tj = P e−tΛP T,
where e−tΛ is the diagonal matrix with elements e−tλj . Recall that for a
symmetric matrix the matrix norm subordinate to the Euclidean norm |v| =
(
∑N
i=1 v
2
i )
1/2 of v ∈ RN we have
|A| = sup
|v|=1
|Av| = max
j
|λj|.
We conclude, since |P | = |P T| = 1, that, with λ1 the smallest eigenvalue of
A,
|E(t)| = |e−tA| = max
j
e−tλj = e−tλ1 .
In particular, if all λj ≥ 0, i.e., if A is positive semidefinite, we find from
(7.7) the stability estimate
|u(t)| ≤ |v| +
∫ t
0
|f (s)| ds, for t ≥ 0.
Similarly, if A is positive definite, so that λ1 > 0, we have
|u(t)| ≤ e−tλ1|v| +
∫ t
0
e−(t−s)λ1|f (s)| ds, for t ≥ 0.
We say that the system (7.5) is stable, or asymptotically stable, in these two
cases, respectively. If A has a negative eigenvalue, however, we have |e−tA| →
∞ as t → ∞, and we then say that the system is unstable.

7.1 The Initial Value Problem for a Linear System 99
In the stable case, the difference between two solutions u1(t) and u2(t)
remains small, if the initial data v1 and v2 and the source terms f1(t) and
f2(t) are close. More precisely, since the difference u1 − u2 is a solution of the
system with right hand side f1 − f2 and initial value v1 − v2, we have
|u1(t) − u2(t)| ≤ |v1 − v2| +
∫ t
0
|f1(s) − f2(s)| ds, for t ≥ 0.
In the asymptotically stable case we have similarly
|u1(t) − u2(t)| = e
−tλ1|v1 − v2| +
∫ t
0
e−(t−s)λ1|f1(s) − f2(s)| ds, for t ≥ 0,
which shows, in particular, that the influence of the initial data and the value
of the source terms at time s decreases exponentially as t → ∞.
The above analysis does not apply if the matrix A in (7.3) depends on t.
To illustrate this we consider the scalar equation
u′ + a(t)u = f (t) for t > 0, with u(0) = v.
Let ã(t) =
∫ t
0
a(s) ds so that ã′(t) = a(t). Then following the same steps as
above, we have
u(t) = e−ã(t)v +
∫ t
0
e−(ã(t)−ã(s))f (s) ds,
but, since in general ã(t) − ã(s) =
∫ t
s
a(τ ) dτ ̸=
∫ t−s
0
a(τ ) dτ = ã(t − s), the
analogue of (7.7) does not hold. Instead, this time we may write
(7.8)
u(t) = E(t, 0)v +
∫ t
0
E(t, s)f (s) ds, with E(t, s) = exp
(
−
∫ t
s
a(τ ) dτ
)
.
For the initial-value problem for the linear system
u′ + A(t)u = f (t), for t > 0,
where A(t) is a matrix, it can be shown that the solution may again be
written as in (7.8), but the matrix E(t, s) will then in general have a more
complicated form. It may be thought of as the operator that takes the value
of the solution of the homogeneous equation u′ + A(t)u = 0 from time s to
time t, so that u(t) = E(t, s)u(s). If A(t) = A is independent of t, then E(t, s)
depends only on the difference t − s and E(t, s) = E(t − s) = e−(t−s)A.
We shall give a glimpse of the general theory for ordinary differential
equations by considering the possibly nonlinear scalar initial-value problem
(7.9) u′ = f (t, u), for t > 0, with u(0) = v,

100 7 Initial-Value Problems for ODEs
where f is now a smooth function of t and u. The equation gives the direction
of the tangent of a solution curve at any point, where the curve is defined
by the points (t, u(t)) ∈ R2. To show that there exists a solution starting at
u(0) = v, i.e., a solution curve u(t) which passes through (0, v), one may use
Euler’s method, which consists in approximating the solution by a polygonal
curve as follows: Let k be a small time step and set tn = nk, n = 0, 1, . . ..
Then the approximation U n to u(tn) is defined successively by
(7.10)
U n − U n−1
k
= f (tn−1, U
n−1), for n ≥ 1,
or
U n = U n−1 + kf (tn−1, U
n−1), for n ≥ 1, with U 0 = v.
This means that starting at the point (tn−1, U
n−1), we follow the tangent
direction defined by the differential equation in (7.9), and take the value at
t = tn as the approximation of u at that point. The approximate solution is
then the continuous, piecewise linear function which takes the value U n at
tn. One may show by that the curves thus defined tend to a limit curve as
k → 0, and that this is our desired solution of (7.9). We refer to a book on
ordinary differential equations for details. Another method for solving (7.9),
Picard’s method, is discussed in Problem 7.4.
We shall now briefly look at second order systems and begin with the
simple scalar problem
(7.11) u′′ + au = 0, for t > 0, with u(0) = v, u′(0) = w,
where a is a positive number. As is well-known, and easily checked, the solu-
tion of this problem is
u(t) = cos(
√
at)v +
1
√
a
sin(
√
at)w, for t ≥ 0.
We next turn to the corresponding system
(7.12) u′′ + Au = 0, for t > 0, with u(0) = v, u′(0) = w,
where now u is an N -vector and A is a symmetric positive definite N × N
matrix. Letting A = P ΛP T we may define
√
A to be the positive definite
matrix P
√
ΛP T, where
√
Λ is the diagonal matrix with the positive square
roots of the eigenvalues of A as its diagonal elements. Note that
√
A has
the same eigenvectors as A. Using the Euler formulas to define cos(B) and
sin(B), where B is an N × N matrix, i.e.,
cos B =
1
2
(
eiB + e−iB
)
, sin B =
1
2i
(
eiB − e−iB
)
,
we then easily find that the solution of (7.12) is

7.2 Numerical Solution of ODEs 101
(7.13) u(t) = cos(t
√
A)v + (
√
A)−1 sin(t
√
A)w, for t ≥ 0.
We note that if {ϕj}
N
j=1 are the normalized eigenvectors of A corresponding
to the eigenvalues {λj}
N
j=1, and if vj = (v, ϕj ) and wj = (w, ϕj ) are the
components of v and w in the direction of ϕj (here (v, w) = v
Tw), then
uj (t) = (u(t), ϕj ) = cos(
√
λj t)vj +
1
√
λj
sin(
√
λj t)wj , for j = 1, . . . , N.
These components thus vary periodically as t grows. In particular, u(t) does
not tend to zero as t tends to ∞, in contrast to the situation for (7.5) with
A symmetric positive definite.
Another way to treat a second order system is to reduce it to first order
by the introduction of a new dependent variable. Thus, in the case of (7.12),
we now set U = (U1, U2)
T = (u, u′)T and obtain the first order system
U ′1 − U2 = 0,
U ′2 + AU1 = 0,
for t > 0, with U (0) =
[
v
w
]
.
The solution is
(7.14) U (t) = exp
(
t
[
0 I
−A 0
])[
v
w
]
, for t ≥ 0.
It is easy to see that this implies (7.13), see Problem 7.7.
7.2 Numerical Solution of ODEs
The Euler method just described may also be used for the numerical solution
of the initial value problem (7.3). Note that even for a system of ordinary
differential equations with constant coefficients we may need numerical meth-
ods, because e−tA may not be very easy to compute if the dimension N is
large.
Let us begin with the model problem
u′ + au = 0, for t > 0, with u(0) = v.
In this case Euler’s method (7.10) gives, for the approximate solution U n at
tn = nk,
U n = (1 − ak)U n−1 = (1 − ak)nv.
(In numerical analysis this method is referred to as the forward Euler method,
since the derivative at tn−1 is replaced by the forward difference quotient
(U n − U n−1)/k.) We find, for t = tn a fixed time, that
U n =
(
1 −
t
n
a
)n
v → e−atv, as n → ∞,

102 7 Initial-Value Problems for ODEs
so that the numerical solution converges to the exact solution as k → 0 in
such a way that nk = t is kept constant.
We shall now discuss the size of the error. Assume a ≥ 0, i.e., that we
are in the stable case for the differential equation. Now take k so small that
1 − ak ≥ −1, or k ≤ 2/a. Then
|U n| = |(1 − ak)nv| ≤ |v|, for n ≥ 0.
so that the numerical solution is also stable. Note that the requirement ak ≤ 2
means that for large a, the time step k has to be chosen ≤ 2/a. If k is larger,
then U n grows with n, in contrast to the behavior of the exact solution of
the differential equation. We have
U n − u(tn) = (1 − ak)
nv − (e−ak)nv
=
(
(1 − ak) − e−ak
) n−1∑
j=0
(1 − ak)j e−(n−1−j)akv.
We find easily by Maclaurin’s formula
|1 − x − e−x| ≤ 1
2
x2, for x ≥ 0,
and hence
|U n − u(tn)| ≤
1
2
a2k2
n−1∑
j=0
|v| = 1
2
nka2k|v| = ( 1
2
tna
2)k|v| = C(tn, a)k|v|,
so that the error is O(k) as k → 0 on any finite interval in time.
Recalling that the previous result is valid only under the stability con-
dition ak ≤ 2, we shall now consider an alternative method which does not
have the latter disadvantage, namely the backward Euler method, in which the
difference quotient is taken in the backward direction, so that U n is defined
by
U n − U n−1
k
+ aU n = 0 for n ≥ 1, with U 0 = v.
This time
U n =
1
1 + ak
U n−1 =
1
(1 + ak)n
v,
and, if a ≥ 0, the stability bound |U n| ≤ |v| holds for n ≥ 0, independently
of the sizes of k and a. Now
(7.15) U n − u(tn) =
( 1
1 + ak
− e−ak
) n−1∑
j=0
1
(1 + ak)j
e−(n−1−j)akv.
Here

7.2 Numerical Solution of ODEs 103
(7.16)
∣
∣
∣
1
1 + x
− e−x
∣
∣
∣ ≤ 2×2, for x ≥ 0,
so that now, without any restriction on k,
|U n − u(tn)| ≤ 2tna
2k |v| = C(tn, a)k|v|.
For numerical purposes it would be desirable to have a higher power of k
than the first in the error bound. This motivates the Crank-Nicolson method,
U n − U n−1
k
+ a
U n + U n−1
2
= 0, for n ≥ 1, with U 0 = v,
which implies
U n =
1 − 1
2
ak
1 + 1
2
ak
U n−1 =
(1 − 1
2
ak
1 + 1
2
ak
)n
v.
Here again, for any k and n, we have the stability property |U n| ≤ |v| for
n ≥ 0. Since
∣
∣
∣
1 − 1
2
x
1 + 1
2
x
− e−x
∣
∣
∣ ≤ x3, for x ≥ 0,
we now have
|U n − u(tn)| =
∣
∣
∣
(1 − 1
2
ak
1 + 1
2
ak
− e−ak
) n−1∑
j=0
(1 − 1
2
ak
1 + 1
2
ak
)j
e−(n−1−j)akv
∣
∣
∣
≤ a3k3
n−1∑
j=0
|v| = tna
3k2|v| = C(tn, a)k
2
|v|.
The error thus tends to zero as O(k2) rather than O(k).
In all the above error estimates the constants on the right grow with a.
We shall now demonstrate that if the backward Euler rule is used, then one
may show an error bound which is independent of a. This is convenient if a
is allowed to become very large. We shall show
(7.17) |U n − u(tn)| ≤ Ct
−1
n k|v|,
where C is independent of a and tn. For fixed tn = t positive, this thus shows
O(k) convergence, uniformly in a. To prove (7.17), consider first ak ≥ 1, say.
Then
(7.18) |U n| =
1
(1 + ak)n
|v| ≤ 2−n|v|.
But, for a suitable C1, we have
2−n ≤ C1/n = C1t
−1
n k.
Further

104 7 Initial-Value Problems for ODEs
|u(tn)| = e
−nak
|v| ≤ e−n|v| ≤ C2n
−1
|v| = C2t
−1
n k|v|.
so that (7.17) holds by the triangle inequality.
In order to treat the case ak ≤ 1, we note that, for suitable γ with
0 < γ ≤ 1, 1 1 + x ≤ e−γx, for 0 ≤ x ≤ 1, so that 1 (1 + ak)j ≤ e−γjak. Hence, using (7.15) and (7.16), we obtain |U n − u(tn)| ≤ 2a 2k2 n−1∑ j=0 e−γjake−γ(n−1−j)ak = 2a2k2ne−γ(n−1)ak ≤ 2eγ a2tne −tnγak ≤ C3t −1 n k, where C3 = 2 e γ sup x≥0 x2e−γx. Together our estimates complete the proof of (7.17). This property is not valid for the Crank-Nicolson method, because the analogue of (7.18) does not hold, since |(1 − 1 2 ak)/(1 + 1 2 ak)| tends to 1 as ak tends to ∞. The strong stability property just described for the backward Euler method is useful when treating systems of the form u′ + Au = f (t), for t > 0,
where A is a symmetric positive definite N ×N matrix, which is not well con-
ditioned, i.e., for which the ratio between the largest and smallest eigenvalues
is large. Such a system is said to be a stiff system of ordinary differential
equations. The backward Euler method in this case is defined by the equa-
tions
(7.19) (I + kA)U n = U n−1 + kf (tn), for n ≥ 1, with U
0 = v,
and we thus have to solve a system of equations at each time step. We there-
fore say that this method is implicit. This is in contrast to the explicit forward
Euler method
(7.20) U n = (I − kA)U n−1 + kf (tn−1), for n ≥ 1, with U
0 = v,
which, however, has the drawbacks concerning stability described above.
The system (7.19) may be written
U n = (I + kA)−1U n−1 + k(I + kA)−1f (tn), for n ≥ 1,
and we note that when A is symmetric positive definite we have

7.2 Numerical Solution of ODEs 105
(7.21) |(I + kA)−1| = max
j
1
1 + kλj
=
1
1 + kλ1
< 1, for any k > 0,
where λ1 is the smallest eigenvalue of A.
Thus, for the homogeneous system, i.e., when f = 0, we have |U n| =
|(I + kA)−nv| → 0 as n → ∞. The numerical solution therefore reproduces
the asymptotic behavior of the differential equations. On the other hand, for
the matrix occurring in (7.20), we have
|I − kA| = max
j
|1 − kλj|,
which is less than 1 only if 1 − kλN > −1, i.e., if k < 2/λN , which could be a very restrictive condition, where λN is the largest eigenvalue of A. In the case of a system the Crank-Nicolson method may be written (I + 1 2 kA)U n = (I − 1 2 kA)U n−1 + kf (tn−1/2), where tn−1/2 = (n − 1/2)k, or U n = (I + 1 2 kA)−1(I − 1 2 kA)U n−1 + k(I + 1 2 kA)−1f (tn−1/2), and it is thus an implicit method. Here |(I + 1 2 kA)−1(I − 1 2 kA)| = max j ∣ ∣ ∣ 1 − 1 2 kλj 1 + 1 2 kλj ∣ ∣ ∣ < 1, for all k. As pointed out previously, for fixed k, the norm tends to 1 if λmax → ∞, which is less satisfactory than (7.21). We close with a short discussion of numerical methods for the initial value problem for the second order scalar equation (7.11). We first replace the second derivative by a symmetric difference quotient and obtain U n+1 − 2U n + U n−1 k2 + aU n = 0, for n ≥ 1, with, for instance, the initial conditions U 0 = v, U 1 − U 0 k = w. The difference equation may also be written (7.22) U n+1 − 2µU n + U n−1 = 0, where µ = 1 − ak2/2, and this difference equation has the characteristic equation τ 2 − 2µτ + 1 = 0. 106 7 Initial-Value Problems for ODEs If its roots τ1,2 are distinct, then the solution of (7.22) is of the form (7.23) U n = c1τ n 1 + c2τ n 2 , with c1 and c2 determined by the initial conditions. For |µ| < 1, i.e., for ak2 < 4, the roots are distinct and |τ1| = |τ2| = 1 so that stability holds, |U n| ≤ C(|v| + |w|), for n ≥ 0. On the other hand, if |µ| > 1, or ak2 > 4, then we have |τ1| > 1 and |τ2| < 1. In this case the general solution of (7.22) increases exponentially. If instead we consider the implicit method, U n+1 − 2U n + U n−1 k2 + a U n+1 = 0, for n ≥ 1, then the characteristic equation becomes ντ 2 − 2τ + 1 = 0, where ν = 1 + ak2. The roots are now less than one in modulus for any choice of k and a and the stability is secured. However, this method is only first order accurate because of the lack of symmetry in the difference approximation. The method U n+1 − 2U n + U n−1 k2 + a (1 4 U n+1 + 1 2 U n + 1 4 U n−1 ) U n+1 = 0, for n ≥ 1, is second order accurate and stable for any k and a because the characteristic equation τ 2 − 2κτ + 1 = 0, where κ = (1 − 1 4 ak2)/(1 + 1 4 ak2). has distinct roots with |τ1| = |τ2| = 1. 7.3 Problems Problem 7.1. Solve the initial value problem u′(t) = [ 1 2 2 1 ] u(t), for t > 0, with u(0) =
[
1
2
]
.
Problem 7.2. (Computer exercise.) Find an approximate solution at t = 1 of
Problem 7.1 by the forward and backward Euler methods and by the Crank-
Nicolson method for k = 1/10, 1/100. Compare with the exact solution.
Problem 7.3. Solve the initial value problem
u′(t) =
[
1 2t
2t 1
]
u(t), for t > 0, with u(0) =
[
1
0
]
.

7.3 Problems 107
Problem 7.4. (Picard’s method.) Prove the existence of solutions to (7.9) as
follows: Show first that a solution of the initial value problem (7.9) satisfies
the integral equation
u(t) = v +
∫ t
0
f (s, u(s)) ds =: T (u)(t),
and, conversely, that a solution of this integral equation is a solution of (7.9).
Assume that the continuous function f (t, u) satisfies a global Lipschitz con-
dition with respect to the second variable, i.e.,
|f (t, v) − f (t, w)| ≤ K|v − w|, ∀v, w ∈ R, 0 ≤ t ≤ a.
Show that the sequence un, n = 0, 1, . . ., defined by
u0(t) = v, un+1(t) = T (un)(t), for n ≥ 0,
satisfies
|un+1(t) − un(t)| ≤ CK
nan+1/(n + 1)!,
that this implies that
∑∞
n=0(un+1(t) − un(t)) converges uniformly to u(t) − v
for 0 ≤ t ≤ a, that u ∈ C([0, a]), and that u = T (u). In particular, this
implies that u satisfies (7.9). Show finally that f (t, u) satisfies a Lipschitz
condition with respect to the second variable if ∂f /∂u is bounded.
Problem 7.5. Prove a uniqueness result for (7.9), when f (t, u) is a continu-
ous function which satisfies a Lipschitz condition with respect to the second
variable (cf. Problem 7.4). Hint: Assume that u1 and u2 are two solutions,
which both satisfy the integral equation of Problem 7.4, and use the Lipschitz
condition to derive an inequality which shows that u1 − u2 = 0.
Problem 7.6. (a) (Gronwall’s lemma.) Suppose that ϕ is a nonnegative con-
tinuous function such that
ϕ(t) ≤ a + b
∫ t
0
ϕ(s) ds, for t > 0,
where a and b are nonnegative constants. Prove that
ϕ(t) ≤ a ebt, for t > 0.
(b) Use Gronwall’s lemma to show that the solution of (7.3) satisfies
|u(t)| ≤ e|A|T
(
|v| +
∫ T
0
|f (s)| ds
)
, for 0 ≤ t ≤ T.
Show that this implies uniqueness of the solution.
Problem 7.7. Show that (7.13) and (7.14) are equivalent.

108 7 Initial-Value Problems for ODEs
Problem 7.8. Prove that the general solution of (7.22) is (7.23), if τ1 ̸= τ2.
Show also that
|µ| < 1 implies |τ1| = |τ2| = 1, |µ| > 1 implies |τ1| < 1, |τ2| > 1.
What is the general form of the solution if τ1 = τ2?

8 Parabolic Equations
In this chapter we study both the pure initial value problem and the mixed
initial-boundary value problem for the model heat equation, using Fourier
techniques as well as energy arguments. In Sect. 8.1 we analyze the solution
of the pure initial value problem for the homogeneous heat equation by means
of a representation in terms of the Gauss kernel, and use it to investigate
properties of the solution. In the remainder of the chapter we consider the
initial-boundary value problem in a bounded spatial domain. In Sect. 8.2 we
solve the homogeneous equation by means of eigenfunction expansions, and
apply Duhamel’s principle to find a solution of the inhomogeneous equation.
In Sect. 8.3 we introduce the variational formulation of the problem and give
examples of the use of energy arguments, and in Sect. 8.4 we show and apply
the maximum principle.
8.1 The Pure Initial Value Problem
We begin our study of parabolic equations by considering the pure initial
value problem (or the Cauchy problem) for the heat equation, which is to
find u(x, t) such that
(8.1)
∂u
∂t
− ∆u = 0, in Rd × R+,
u(·, 0) = v, in Rd.
We shall employ the Fourier transform of u with respect to x, cf. App. A.3,
û(ξ, t) = Fu(·, t)(ξ) =
∫
Rd
u(x, t)e−ix·ξ dx, for ξ ∈ Rd.
If u and its derivatives are small enough for large |x|, then we have
(
F∆u(·, t)
)
(ξ) =
∫
Rd
∆u(x, t)e−ix·ξ dx = −|ξ|2û(ξ, t)
and, with ut = ∂u/∂t,
(
Fut(·, t)
)
(ξ) =
dû
dt
(ξ, t).

110 8 Parabolic Equations
Hence we conclude from (8.1) that û satisfies
dû
dt
= −|ξ|2û, for ξ ∈ Rd, t > 0,
û(ξ, 0) = v̂(ξ), for ξ ∈ Rd.
This is a simple initial value problem for a first order linear ordinary differ-
ential equation, with ξ as a parameter, and its solution is
(8.2) û(ξ, t) = v̂(ξ)e−t|ξ|
2
.
Recalling that w(x) = e−|x|
2
has the Fourier transform
ŵ(ξ) = πd/2e−|ξ|
2/4
(cf. Problem A.19), we conclude from (A.34) that e−t|ξ|
2
is the Fourier trans-
form of the Gauss kernel
U (x, t) = (4πt)−d/2e−|x|
2/4t,
and hence we obtain formally from (8.2) that
(8.3) u(x, t) =
(
U (·, t) ∗ v
)
(x) = (4πt)−d/2
∫
Rd
v(y)e−|x−y|
2/4t dy.
The function U (x, t) is a fundamental solution of the initial value problem.
We shall now show that the function defined in (8.3) is, in fact, a solution of
(8.1) under a weak assumption on the initial function. Note that U (x, t) and
u(x, t) in (8.3) are only defined for t > 0.
Theorem 8.1. If v is a bounded continuous function on Rd, then the func-
tion u(x, t) defined by (8.3) is a solution of the heat equation for t > 0, and
tends to the initial data v as t tends to 0.
Proof. We first note that for t > 0 we may differentiate the integral in (8.3)
with respect to x and t under the integral sign, and show directly that this
function satisfies the heat equation in (8.1). To see that u(x, t) tends to the
desired initial values as t → 0 we let x0 ∈ R
d be arbitrary and show that
u(x, t) → v(x0), as (x, t) → (x0, 0).
In fact, using the transformation η = (y − x)/
√
4t, and the formula
(8.4) π−d/2
∫
Rd
e−|x|
2
dx = 1,
we may write

8.1 The Pure Initial Value Problem 111
u(x, t) − v(x0) = (4πt)
−d/2
∫
Rd
v(y)e−|x−y|
2/4t dy − v(x0)
= π−d/2
∫
Rd
(
v(x +
√
4tη) − v(x0)
)
e−|η|
2
dη.
Let ϵ > 0 be arbitrary and let δ be so small that
(8.5) |v(z) − v(x0)| < ϵ, if |z − x0| < δ. Let M = ∥v∥C = ∥v∥C(Rd). For any ω > 0, we have
|u(x, t) − v(x0)| ≤ 2M π
−d/2
∫
|y|>ω
e−|y|
2
dy
+ π−d/2
∫
|y|<ω ∣ ∣v(x + √ 4ty) − v(x0) ∣ ∣ e−|y| 2 dy = I + II. We now fix ω so large that I < ϵ, which is possible in view of (8.4). Then, with ω fixed, we obtain, using (8.5) and (8.4), II ≤ sup |y|<ω |v(x + √ 4ty) − v(x0)| < ϵ, if |x − x0| + √ 4tω < δ. Hence, for these x, t we have |u(x, t) − v(x0)| < 2ϵ, which completes the proof. ⊓. Theorem 8.1 thus shows that the initial value problem (8.1) admits a solu- tion, and is therefore an existence theorem. We shall show that this solution depends continuously on the initial data v. We write (8.3) in the form (8.6) u(x, t) = (E(t)v)(x) = (4πt)−d/2 ∫ Rd v(y)e−|x−y| 2/4t dy, where we may think of E(t) as defining a linear operator, the solution operator of (8.1), which takes the given initial data into the solution at time t. Note that by (8.4), with ∥v∥C = ∥v∥C(Rd), |u(x, t)| ≤ (4πt)−d/2 ∫ Rd e−|x−y| 2/4t dy ∥v∥C = ∥v∥C, so that ∥u(·, t)∥C ≤ ∥v∥C, for t > 0.
This shows that the operator E(t) is bounded with respect to the maximum-
norm, with operator norm 1, which is the first part of the following result.

112 8 Parabolic Equations
Theorem 8.2. The solution operator E(t) defined by (8.6) is bounded in C,
and
(8.7) ∥E(t)v∥C ≤ ∥v∥C, for t ≥ 0.
If v1 and v2 are two bounded continuous functions on R
d and u1 and u2 are
the corresponding solutions of the initial value problem (8.1), then
(8.8) ∥u1(t) − u2(t)∥C ≤ ∥v1 − v2∥C, for t ≥ 0.
Proof. It remains only to show the second part of the theorem. But, since
E(t) is a linear operator,
u1(t) − u2(t) = E(t)v1 − E(t)v2 = E(t)(v1 − v2),
and hence (8.8) follows at once from (8.7). ⊓.
By using a maximum principle we shall prove in Sect. 8.4 the correspond-
ing uniqueness result, i.e., that there exists at most one bounded solution of
(8.1) and thus (8.3) is the only one.
Together the existence, uniqueness, and continuous dependence properties
make the problem (8.1) a well posed problem. In particular, the continuous de-
pendence property is important in applications. It shows that a small change
in the data of the problem has only a small effect on the solution.
Not all problems which admit solutions have this continuous dependence
property. Consider for example the initial value problem
(8.9)
ut + uxx = 0, in R × R+,
u(x, 0) = vn(x) = n
−1 sin(nx), for x ∈ R,
which has the solution
un(x, t) = n
−1en
2t sin(nx).
Here
∥vn∥C = n
−1
→ 0, as n → ∞,
whereas, for any t > 0,
∥un(t)∥C = n
−1en
2t
→ ∞, as n → ∞.
Hence, although the initial value vn is close to 0, the solution is not for t > 0.
The differential equation in (8.9) is the heat equation with the sign for the
time derivative reversed, i.e., the backward heat equation. The result therefore
means that the problem of determining an earlier distribution of heat in a
body from the present one is ill posed.
We have already noted above that the representation of u(x, t) in terms
of v in (8.3) allows differentiation with respect to both x and t under the

8.1 The Pure Initial Value Problem 113
integral sign for t > 0, even without regularity assumptions on v. In fact, this
differentiation can be carried out an arbitrary number of times so that u is
infinitely differentiable, u ∈ C∞, for t > 0. Using the multi-index notation
from (1.8), one finds easily
|D
j
t D
αU (x, t)| ≤ t−j−|α|/2−d/2P (|x|/
√
4t)e−|x|
2/4t
≤ Ct−j−|α|/2−d/2e−|x|
2/8t,
where P (y) is a polynomial in y, and where we have used the fact that for
any polynomial P there is a C such that
|P (y)e−y
2
| ≤ Ce−y
2/2, for y > 0.
Hence
sup
x∈Rd
|D
j
t D
αu(x, t)| ≤ Ct−j−|α|/2−d/2 sup
x∈Rd
∫
Rd
|v(y)|e−|x−y|
2/8t dy
≤ Ct−j−|α|/2 sup
y∈Rd
|v(y)|,
or
∥D
j
t D
αE(t)v∥C ≤ Ct
−j−|α|/2
∥v∥C, for t > 0,
which shows that the operator E(t) has a smoothing property: The solution
of (8.1) is smooth for t > 0 even if v is nonsmooth. However, the bounds for
the derivatives then grow as t tends to zero.
On the other hand, if the initial values are smooth then the derivatives of
the solution are bounded uniformly down to t = 0: We have from (8.6), after
the change of variables z = x − y,
(
DαE(t)v
)
(x) = Dαx u(x, t) = (4πt)
−d/2Dαx
∫
Rd
v(x − z)e−|z|
2/4t dz
= (4πt)−d/2
∫
Rd
Dαx v(x − z)e
−|z|2/4t dz = (E(t)Dαv)(x),
and hence, by (8.1) and (8.7),
∥D
j
t D
αE(t)v∥C = ∥∆
j DαE(t)v∥C = ∥E(t)∆
j Dαv∥C ≤ ∥∆
j Dαv∥C.
It can be shown that the solution of the initial value problem for the
inhomogeneous heat equation,
ut − ∆u = f, in R
d
× R+,
u(·, 0) = v, in Rd,
where f = f (x, t) is given, may be represented in the form

114 8 Parabolic Equations
u(x, t) =
∫
Rd
v(y)U (x − y, t) dy +
∫ t
0
∫
Rd
f (y, s)U (x − y, t − s) dy ds
= E(t)v +
∫ t
0
E(t − s)f (·, s) ds,
provided, e.g., that v, f , and ∇f are continuous and bounded.
8.2 Solution of the Initial-Boundary Value Problem by
Eigenfunction Expansion
We shall first consider the mixed initial-boundary value problem for the ho-
mogeneous heat equation: Find u(x, t) such that
(8.10)
ut − ∆u = 0 in Ω × R+,
u = 0, on Γ × R+,
u(·, 0) = v in Ω,
where Ω is a bounded domain in Rd with smooth boundary Γ , ut = ∂u/∂t,
and v is a given function in L2 = L2(Ω). We shall now solve this problem by
using eigenfunction expansions. We denote by (·, ·) and ∥·∥ the inner product
and norm in L2 = L2(Ω), respectively.
We recall from Chapt. 6 that there exists an orthonormal basis {ϕi}
∞
i=1
in L2 of smooth eigenfunctions ϕi and corresponding eigenvalues {λi}
∞
i=1
satisfying
(8.11) −∆ϕi = λiϕi in Ω, with ϕi = 0 on Γ,
or, equivalently, with our usual notation
a(ϕi, v) =
∫
Ω
∇ϕi · ∇v dx = λi(ϕi, v), ∀v ∈ H
1
0 ,
Recall that 0 < λ1 < λ2 ≤ · · · ≤ λi ≤ · · · , that λi → ∞ as i → ∞, and that (with Kronecker’s symbol δij = 1 for j = i and 0 otherwise) a(ϕi, ϕj ) = λiδij . We now seek a solution to (8.10) of the form (8.12) u(x, t) = ∞∑ i=1 ûi(t)ϕi(x), where the ûi : R+ → R are coefficients to be determined. Because this is a sum of products of functions of x and t this approach is also called the method of separation of variables. Inserting (8.12) into the differential equation in (8.10) and using (8.11) we obtain formally 8.2 Solution by Eigenfunction Expansion 115 ∞∑ i=1 ( û′i(t) + λiûi(t) ) ϕi(x) = 0, for x ∈ Ω, t ∈ R+, and hence, since the ϕi form a basis, û′i(t) + λiûi(t) = 0, for t ∈ R+, i = 1, 2, . . . , so that ûi(t) = ûi(0)e −λit. Moreover, from the initial condition in (8.10) it follows that u(·, 0) = ∞∑ i=1 ûi(0)ϕi = v = ∞∑ i=1 v̂iϕi, where v̂i = (v, ϕi) = ∫ Ω v ϕi dx. We thus see that, at least formally, the solution of (8.10) has to be (8.13) u(x, t) = ∞∑ i=1 v̂ie −λitϕi(x), where by Parseval’s relation, with ∥ · ∥ = ∥ · ∥L2 , ∥u(·, t)∥2 = ∞∑ i=1 ( v̂ie −λit )2 ≤ e−2λ1t ∞∑ i=1 v̂2i = e −2λ1t∥v∥2 < ∞, Thus u(·, t) ∈ L2 for t ≥ 0, and its L2-norm decreases exponentially as t → ∞. Although this decay is important in some situations, for simplicity we shall refrain from keeping track of the behavior of u(·, t) for large t in the sequel and content ourselves with the conclusion that ∥u(·, t)∥ ≤ ∥v∥, for t ∈ R+. We now show that for t > 0 the function u(·, t) defined in (8.13) is smooth
and satisfies the differential equation and the boundary condition in (8.10)
in the classical sense, and that the initial condition holds in the sense that
(8.14) ∥u(·, t) − v∥ → 0, as t → 0.
We first note that for any k ≥ 0 there is a constant Ck such that s
ke−s ≤
Ck for s ≥ 0, Using this with k = 1 we have
|u(·, t)|21 =
∞∑
i=1
λi
(
v̂ie
−λit
)2
= t−1
∞∑
i=1
v̂2i (λit) e
−2λit ≤ C1t
−1
∥v∥2,
so that
(8.15) |u(·, t)|1 ≤ Ct
−1/2
∥v∥, for t > 0.

116 8 Parabolic Equations
Thus u(·, t) ∈ H10 for t > 0, by Theorem 6.4, and, in particular, u(·, t) satisfies
the boundary condition in (8.10). Now, applying (−∆)k to each term in
(8.13), we obtain since −∆ϕi = λiϕi
(8.16) (−∆)ku(x, t) =
∞∑
i=1
v̂iλ
k
i e
−λitϕi(x),
and hence, for t > 0,
∥∆ku(·, t)∥2 =
∞∑
i=1
(
v̂iλ
k
i e
−λit
)2
≤ C2k t
−2k
∞∑
i=1
v̂2i = C
2
k t
−2k
∥v∥2 < ∞. In the same way as in (8.15), we also have |∆ku(·, t)|1 ≤ Ckt −k−1/2 ∥v∥ < ∞, for t > 0,
and thus ∆ku(·, t) = 0 on Γ for any k ≥ 0 when t > 0. We may also apply
Dmt to each term in (8.16), and since Dte
−λit = −λie
−λit, we obtain
|Dmt ∆
ku(·, t)|δ ≤ Ct
−m−k−δ/2
∥v∥ < ∞, for t > 0, δ = 0, 1.
Recall from the theory of elliptic equations the regularity estimate (3.37),
∥w∥s ≤ C∥∆w∥s−2, ∀w ∈ H
s
∩ H10 , for s ≥ 2.
By repeated application of this we obtain, again for δ = 0 or 1,
∥w∥2k+δ ≤ C∥∆
kw∥δ, ∀w ∈ H
2k+δ, if ∆j w = 0 on Γ for j < k, and we finally conclude that, for any nonnegative integers s and m, (8.17) ∥Dmt u(·, t)∥s ≤ Ct −m−s/2 ∥v∥, for t > 0.
It follows by Sobolev’s inequality, Theorem A.5, that Dmt u(·, t) ∈ C
p for t > 0,
for any p ≥ 0.
Thus u(x, t) is a smooth function of x and t for t > 0 even though we have
not assumed the initial data v to be smooth, and u(·, t) therefore satisfies the
heat equation in the classical sense. By above we also know that the boundary
condition is satisfied, and finally we obtain (8.14) by showing that
∥u(·, t) − v∥2 =
∞∑
i=1
(
e−λit − 1
)2
v̂2i → 0, as t → 0.
To prove this we let ϵ > 0 be arbitrarily small and choose N large enough
that
∑∞
i=N+1 v̂
2
i < ϵ . Then ∥u(·, t) − v∥2 ≤ N∑ i=1 ( e−λit − 1 )2 v̂2i + ϵ. 8.2 Solution by Eigenfunction Expansion 117 Since each of the terms of the sum tends to zero as t → 0, we conclude that ∥u(·, t) − v∥2 < 2ϵ, for t small enough. We collect these results in the following theorem. Theorem 8.3. For any v ∈ L2 the function u(x, t) defined by (8.13) is a classical solution of the heat equation in (8.10), vanishes on Γ for t > 0, and
satisfies the initial condition in the sense of (8.14). Moreover, the smoothness
estimate (8.17) holds.
Since the factor t−m−s/2 on the right in (8.17) tends to infinity as t tends
to zero, the smoothness of the solution is not guaranteed uniformly down to
t = 0. If the initial function is smoother, then better results are possible in
this regard. We have, for instance, the following result in H10 .
Theorem 8.4. Assume that v ∈ H10 . Then the solution u(x, t) of (8.10)
determined in Theorem 8.3 satisfies
|u(·, t)|1 ≤ |v|1 for t ≥ 0.
Proof. We have by Theorem 6.4
|u(·, t)|21 =
∞∑
i=1
λiv̂
2
i e
−2λit ≤
∞∑
i=1
λiv̂
2
i = |v|
2
1,
which shows our claim. ⊓.
We note that this result requires not only that the initial data are in H1
but also that they vanish on Γ . This means that the initial data have to be
compatible with the boundary data on Γ × R+, which is obviously required
for the solution to be continuous at t = 0. For higher order regularity further
compatibility conditions are needed.
In the same way as in Sect. 8.1 we may think of the solution at time t as
the result of a solution operator E(t) acting on the initial data v, and thus
write u(t) = E(t)v. By (8.13) this operator satisfies the stability estimate
∥E(t)v∥ ≤ ∥v∥, for t > 0,
and the estimate (8.17) may be expressed as
(8.18) ∥Dmt E(t)v∥s ≤ Ct
−m−s/2
∥v∥, for t > 0, m, s ≥ 0,
which expresses a smoothing property of the solution operator.
The following simple example illustrates the above solution method.

118 8 Parabolic Equations
Example 8.1. The solution of the spatially one-dimensional problem
(8.19)
ut − uxx = 0, in Ω × R+,
u(0, ·) = u(π, ·) = 0, in R+,
u(·, 0) = v, in Ω,
with Ω = (0, π) and v ∈ L2(Ω), is given by
(8.20) u(x, t) =
∞∑
j=1
v̂j e
−j2t sin(jx), where v̂j =
2
π
∫ π
0
v(x) sin(jx) dx.
In this case the associated eigenvalue problem (8.11) reduces to (6.22) with
b = π, and the result thus follows from Theorem 8.3 and the results obtained
in Sect. 6.1, except that the eigenfunctions are not normalized here. Note
that in (8.20) the coefficient v̂j e
−j2t of the eigenfunction sin(jx) is obtained
by multiplying the corresponding coefficient v̂j in the expansion of the initial
function v by the factor e−j
2t. If j is large, then sin(jx) is rapidly oscillating
and the factor e−j
2t rapidly becomes very small as t increases from 0. Thus,
the components of the solution u(x, t) corresponding to the eigenfunctions
sin(jx) with j large are strongly damped as t grows. This means that rapid
variations or oscillations in the initial function v, such as, for instance, in the
case of a discontinuity (jump), are smoothed out as t increases. This is thus
a special case of the smoothing property of the solution operator discussed
above, which is typical for parabolic problems.
The solution operator E(t) introduced above is convenient to use in the
study of the boundary value problem for the inhomogeneous equation,
(8.21)
ut − ∆u = f, in Ω × R+,
u = 0, on Γ × R+,
u(·, 0) = v, in Ω,
In fact, as we shall see, the solution of this problem may be expressed as
(8.22) u(t) = E(t)v +
∫ t
0
E(t − s)f (s) ds,
where we write u(t) for u(·, t) and similarly for f (s). This formula represents
the solution of the inhomogeneous equation as a superposition of solutions of
homogeneous equations, and is referred to as Duhamel’s principle.
Clearly, since E(t) is bounded in L2-norm, the right hand side of (8.22) is
well defined. The first term is the solution of (8.1), the second term vanishes
for t = 0, and both terms vanish on Γ ×R+. Therefore, in order to show that
u in (8.21) is a solution of (8.22), we need to demonstrate that

8.2 Solution by Eigenfunction Expansion 119
(8.23) DtF (t) − ∆F (t) = f (t), where F (t) =
∫ t
0
E(t − s)f (s) ds.
Formally, by differentiation of the integral, we have
(8.24) DtF (t)−∆F (t) = f (t)+
∫ t
0
DtE(t−s)f (s) ds−
∫ t
0
∆E(t−s)f (s) ds,
and since DtE(t − s) = ∆E(t − s) the integrals should cancel. However, if
we require only f (s) ∈ L2 for s ∈ (0, t), then (8.18) indicates a singularity of
order O((t − s)−1) in the integrands, so that the integrals are not necessarily
well defined. For this reason we now assume that ∥Dtf (t)∥ is bounded for
t ∈ [0, T ] with arbitrary T > 0 , and write, after replacing t − s by s in the
last term,
F (t) =
∫ t
0
E(t − s)(f (s) − f (t)) ds +
∫ t
0
E(s)f (t) ds.
By differentiation with respect to t we obtain
(8.25) DtF (t) =
∫ t
0
DtE(t − s)(f (s) − f (t)) ds + E(t)f (t),
where the integrand is now bounded since ∥f (s) − f (t)∥ ≤ C|s − t|. Similarly,
since ∆E(t − s) = DtE(t − s),
(8.26) ∆F (t) =
∫ t
0
∆E(t − s)(f (s) − f (t)) ds + (E(t) − I)f (t).
Taking the difference between (8.25) and (8.26) now shows (8.23).
Another way to deal with the singularities in the integrands in (8.24)
would be to use regularity of f (s) in the spatial variable, e.g., through the
inequality ∥∆E(t − s)f (s)∥ ≤ ∥∆f (s)∥. However, in addition to regularity of
f (s) this would require the unnatural boundary condition f (s) = 0 on Γ .
By (8.22) we obtain at once the stability estimate
(8.27) ∥u(t)∥ ≤ ∥v∥ +
∫ t
0
∥f (s)∥ ds.
In the standard way this may be used to show uniqueness of the solution of
(8.21) as well as the continuous dependence of the solution on the data. For
example, if u1 and u2 are solutions corresponding to the right-hand sides f1
and f2 and initial values v1 and v2, then we have
(8.28) ∥u1(t) − u2(t)∥ ≤ ∥v1 − v2∥ +
∫ t
0
∥f1(s) − f2(s)∥ ds, for t ∈ R+.

120 8 Parabolic Equations
8.3 Variational Formulation. Energy Estimates
We shall now write the initial-boundary value problem (8.21) in variational, or
weak form, and use this to derive some estimates for its solution. Although we
shall not pursue this here, variational methods may be used to prove existence
and uniqueness of solutions of parabolic problems which are considerably
more general than (8.21), such as problems with time-dependent coefficients
or non-selfadjoint elliptic operator, problems with inhomogeneous boundary
conditions, and also some nonlinear problems. For such problems the method
of eigenfunction expansion of the previous section is difficult or impossible to
use. Moreover, the variational formulation is the basis for the finite element
method for parabolic problems, which we shall study in Chapt. 10.
For the variational formulation we multiply the heat equation in (8.21) by
a smooth function ϕ = ϕ(x), which vanishes on Γ and find, after integration
over Ω and using Green’s formula, that
(8.29) (ut, ϕ) + a(u, ϕ) = (f, ϕ), ∀ϕ ∈ H
1
0 , t ∈ R+,
with our standard notation
a(v, w) =
∫
Ω
∇v · ∇w dx, (v, w) =
∫
Ω
vw dx.
The variational problem may then be formulated: Find u = u(x, t) ∈ H10 ,
thus vanishing on Γ , for t > 0, such that (8.29) holds and such that
(8.30) u(·, 0) = v in Ω.
By taking the above steps in the opposite order it is easy to see that if u
is a sufficiently smooth solution of this problem, it is also a solution of (8.21).
In fact, by integration by parts in (8.29) we obtain
(ut − ∆u − f, ϕ) = 0, ∀ϕ ∈ H
1
0 , t ∈ R+,
or, for any t ∈ R+,
∫
Ω
ρ(·, t) ϕ dx = 0, ∀ϕ ∈ H10 , where ρ = ut − ∆u − f.
We conclude, in the same way as for the stationary problem, that this is
possible only if ρ = 0.
The following result shows some bounds in various natural norms for the
solution of our above problem in terms of its data. We proceed formally and
refrain from precise statements about the regularity requirements needed. We
write u(t) for u(·, t) and similarly for f (t).
Theorem 8.5. Let u(t) satisfy (8.29) and (8.30), vanish on Γ , and be ap-
propriately smooth for t ≥ 0. Then there is a constant C such that, for t ≥ 0,

8.3 Variational Formulation. Energy Estimates 121
(8.31) ∥u(t)∥2 +
∫ t
0
|u(s)|21 ds ≤ ∥v∥
2 + C
∫ t
0
∥f (s)∥2 ds
and
(8.32) |u(t)|21 +
∫ t
0
∥ut(s)∥
2 ds ≤ |v|21 +
∫ t
0
∥f (s)∥2 ds.
Proof. Taking ϕ = u in (8.29) we obtain
(8.33) (ut, u) + a(u, u) = (f, u), for t > 0.
Here
(ut, u) =
∫
Ω
utu dx =
∫
Ω
1
2
(u2)t dx =
1
2
d
dt
∥u∥2.
Applying Poincaré’s inequality, Theorem A.6, i.e.,
∥ϕ∥ ≤ C|ϕ|1, for ϕ ∈ H
1
0 ,
we have, using also the inequality 2ab ≤ a2 + b2, that
|(f, u)| ≤ ∥f∥ ∥u∥ ≤ C∥f∥ |u|1 ≤
1
2
|u|21 +
1
2
C2∥f∥2.
Recalling a(u, u) = |u|21, we thus obtain from (8.33) that
1
2
d
dt
∥u∥2 + |u|21 ≤
1
2
|u|21 +
1
2
C2∥f∥2,
or, with a new C,
d
dt
∥u∥2 + |u|21 ≤ C∥f∥
2.
By integration over (0, t) this yields
∥u(t)∥2 +
∫ t
0
|u(s)|21 ds ≤ ∥v∥
2 + C
∫ t
0
∥f∥2 ds,
which is (8.31).
To prove (8.32) we now choose ϕ = ut in (8.29) and obtain
∥ut∥
2 + a(u, ut) = (f, ut) ≤
1
2
∥f∥2 + 1
2
∥ut∥
2.
Here
a(u, ut) =
∫
Ω
∇u · ∇ut dx =
∫
Ω
1
2
(|∇u|2)t dx =
1
2
d
dt
|u|21,
so that we may conclude,
∥ut∥
2 +
d
dt
|u|21 ≤ ∥f∥
2,
whence, by integration over (0, t),
|u(t)|21 +
∫ t
0
∥ut∥
2 ds ≤ |v|21 +
∫ t
0
∥f∥2 ds,
which is (8.32). ⊓.

122 8 Parabolic Equations
It follows in the standard way from (8.31) that if u1 and u2 are solutions
corresponding to the right-hand sides f1 and f2 and initial values v1 and v2,
then we have
∥u1(t)−u2(t)∥
2+
∫ t
0
|u1−u2|
2
1 ds ≤ ∥v1−v2∥
2+C
∫ t
0
∥f1−f2∥
2 ds, for t ≥ 0,
and a similar bound is obtained from (8.32). Note that these estimates also
bound the error in H10 and uses the L2-norm in time rather than the L1-norm
employed in (8.28).
8.4 A Maximum Principle
We now consider the generalization of the mixed initial-boundary value prob-
lem of Sect. 8.2 which allows a source term and inhomogeneous boundary
conditions, i.e., to find u on Ω̄ × Ī such that
(8.34)
ut − ∆u = f, in Ω × I,
u = g, on Γ × I,
u(·, 0) = v, in Ω,
where Ω is a bounded domain in Rd and I = (0, T ) is a finite interval in time.
In order to show a maximum principle for this problem it is convenient to
introduce the parabolic boundary of Ω × I as the set Γp = (Γ × Ī) ∪ (Ω ×{t =
0}), i.e., the boundary of Ω × I minus the interior of the top part of this
boundary, Ω × {t = T }.
Theorem 8.6. Let u be smooth and assume that ut −∆u ≤ 0 in Ω ×I. Then
u attains its maximum on the parabolic boundary Γp.
Proof. If this were not true, then the maximum would be attained either at
an interior point of Ω × I or at a point of Ω × {t = T }, i.e., at a point
(x̄, t̄) ∈ Ω × (0, T ], and we would have
u(x̄, t̄) = max
Ω̄×Ī
u = M > m = max
Γp
u.
In such a case, for ϵ > 0 sufficiently small, the function
w(x, t) = u(x, t) + ϵ|x|2
would also take its maximum at a point in Ω × (0, T ], since, for ϵ small,
max
Γp
w ≤ m + ϵ max
Γp
|x|2 < M ≤ max Ω̄×Ī w. By our assumption we have since ∆(|x|2) = 2d that 8.4 A Maximum Principle 123 (8.35) wt − ∆w = ut − ∆u − 2dϵ < 0, in Ω × I. On the other hand, at the point (x̃, t̃), where w takes its maximum, we have −∆w(x̃, t̃) = − d∑ i=1 wxixi (x̃, t̃) ≥ 0, and wt(x̃, t̃) = 0, if t̃ < T, or wt(x̃, t̃) ≥ 0, if t̃ = T, so that in both cases wt(x̃, t̃) − ∆w(x̃, t̃) ≥ 0. This is a contradiction to (8.35) and thus shows our claim. ⊓. By considering the functions ±u, it follows, in particular, that a solution of the homogeneous heat equation (f = 0) attains both its maximum and its minimum on Γp, so that in this case, with ∥w∥C(M̄) = maxx∈M̄ |w(x)|, ∥u∥C(Ω̄×Ī) ≤ max { ∥g∥C(Γ ×Ī), ∥v∥C(Ω̄) } . For the inhomogeneous equation one may show the following inequality, the proof of which we leave as an exercise, see Problem 8.7. Theorem 8.7. The solution of (8.34) satisfies ∥u∥C(Ω̄×Ī) ≤ max { ∥g∥C(Γ ×Ī), ∥v∥C(Ω̄) } + r2 2d ∥f∥C(Ω̄×Ī), where r is the radius of a ball containing Ω. As usual such a result shows uniqueness and stability for the initial- boundary value problem. We close this section by proving the uniqueness of a bounded solution to the pure initial value problem considered in Sect. 8.1. Theorem 8.8. The initial value problem (8.1) has at most one solution which is bounded in Rd × [0, T ], where T is arbitrary. Proof. If there were two solutions of (8.1), then their difference would be a solution with initial data zero. It suffices therefore to show that the only bounded solution u of ut = ∆u, in R d × I, where I = (0, T ), u(·, 0) = 0, in Rd, is u = 0, or that and if (x0, t0) is an arbitrary point in R d × I, and ϵ > 0 is
arbitrary, then |u(x0, t0)| ≤ ϵ. We introduce the auxiliary function

124 8 Parabolic Equations
w(x, t) =
|x|2 + 2d t
|x0|2 + 2d t0
,
and note that wt = ∆w. Let now
h±(x, t) = −ϵw(x, t) ± u(x, t).
Then
(h±)t − ∆h± = 0, in R
d
× I.
Since u is bounded we have |u(x, t)| ≤ M on Rd × I for some M . Defining R
by R2 = max
(
|x0|
2, M (|x0|
2 + 2dt0)/ϵ
)
, we have
h±(x, t) ≤ −ϵ
R2
|x0|2 + 2d t0
+ M ≤ 0, if |x| = R,
and
h±(x, 0) = −ϵ|x|
2/(|x0|
2 + 2d t0) ≤ 0, for x ∈ R
d.
Hence we may apply Theorem 8.6 with Ω = {|x| < R} and conclude that h±(x, t) ≤ 0 for (x, t) ∈ Ω × I. In particular, at (x0, t0) we have ±u(x0, t0) = h±(x0, t0) + ϵ ≤ ϵ, which completes the proof of the theorem. ⊓. The assumption of Theorem 8.8 that the solutions are bounded in Rd × [0, T ] may be relaxed to the requirement that |u(x, t)| ≤ M ec|x| 2 for all x ∈ Rd, 0 ≤ t ≤ T, and for some M, c > 0, but without some such restriction
on the growth of the solution for large |x|, uniqueness is not guaranteed.
For instance, the following function is a solution of the homogeneous heat
equation which has initial values zero but does not vanish identically for
t > 0:
u(x, t) =
∞∑
n=0
f (n)(t)
x2n
(2n)!
, where f (t) = e−1/t
2
for t > 0, f (0) = 0.
The technical part of the proof is to show that the series converges so rapidly
that it may be differentiated termwise. Then it is obvious that ut = uxx and
that u(x, 0) = 0.
8.5 Problems
Problem 8.1. Show that if u is a solution of (8.1) with
∫
Rd
|v(x)| dx < ∞, then ∫ Rd u(x, t) dx = constant = ∫ Rd v(x) dx, for t ≥ 0. Give a physical interpretation of this result. 8.5 Problems 125 Problem 8.2. Find the solution of the initial-boundary value problem (8.19) with v(x) = 1, for 0 < x < π;(a) v(x) = x(π − x), for 0 < x < π.(b) Sketch the solutions u(x, t) at various time levels t. Problem 8.3. Consider the function u(x, t) = { xt−3/2e−x 2/4t, for t > 0,
0, for t = 0.
Show that u is a solution of
ut − uxx = 0, in R × R+,
and that, for each x,
u(x, t) → 0, as t → 0.
Why is this not a counter-example to the uniqueness result of Theorem 8.8?
Hint: set x = t.
Problem 8.4. Let u = E(t)v be the solution of (8.10). Show that E has the
semigroup property (7.1). Show the estimates
∥u(t)∥ ≤ e−λ1t∥v∥, for t ≥ 0,(a)
∥∆kD
j
t u(t)∥ ≤ Ct
−(j+k)e−λ1t/2∥v∥, for t > 0.(b)
Problem 8.5. Prove by the energy method that there is a constant C =
C(T ) such that if u satisfies (8.29) and (8.30), then
∫ t
0
s∥ut(s)∥
2 ds ≤ C
(
∥v∥2 +
∫ t
0
∥f (s)∥2 ds
)
, for 0 ≤ t ≤ T,(a)
|u(t)|21 ≤ Ct
−1
(
∥v∥2 +
∫ t
0
∥f (s)∥2 ds
)
, for 0 < t ≤ T.(b) Problem 8.6. Let u be the solution of (8.29) and (8.30) with v = 0. Show that ∫ t 0 ( ∥ut(s)∥ 2 + ∥∆u(s)∥2 ) ds ≤ C ∫ t 0 ∥f (s)∥2 ds, for t ≥ 0. Problem 8.7. Prove Theorem 8.7. Hint: See the proof of Theorem 3.2. Problem 8.8. Show estimates analogous to those of Theorem 8.5 when the term −∆u in (8.21) is replaced by Au = −∇ · (a∇u) + b · ∇u + cu as in Sect. 3.5. 126 8 Parabolic Equations Problem 8.9. Prove (8.4). Problem 8.10. Show that if u satisfies (8.10), then there is a constant C such that ∥u(t)∥22 + ∫ t 0 |ut(s)| 2 1 ds ≤ C∥v∥ 2 2, ∀v ∈ H 2 ∩ H10 , t ≥ 0. You can use either the spectral method or the energy method. You also need the elliptic regularity estimate (3.36). Problem 8.11. Let u be the solution of ut − ∆u = 0, in Ω × R+, u(x, t) = 0, on Γ × R+, u(·, 0) = v, in Ω, where Ω = {x ∈ R2 : 0 < xi < 1, i = 1, 2}. Let ϕ(x) = A sin(πx1) sin(πx2) with A > 0. Show that if 0 ≤ v(x) ≤ ϕ(x) for x ∈ Ω, then 0 ≤ u(x, t) ≤
e−2π
2tϕ(x) for x ∈ Ω, t > 0. Hint: Use the maximum principle.
Problem 8.12. Prove the L2 version of Theorem 8.1: If v ∈ L2(R
d) then
∥E(t)v∥L2 ≤ ∥v∥L2 for t ≥ 0 and ∥E(t)v−v∥L2 → 0 as t → 0. Hint: Parseval’s
formula (A.32).
Problem 8.13. Consider the heat equation with Neumann’s boundary con-
dition:
ut − ∆u = 0, in Ω × R+,
∂u
∂n
= 0, on Γ × R+,
u(·, 0) = v, in Ω,
where ∂u/∂n is the outward normal derivative.
(a) Show that u(t) = v for t ≥ 0, where v = 1|Ω|
∫
Ω
v(x) dx denotes the
average value of v.
(b) Show that ∥u(t) − v∥ → 0 as t → ∞.
Problem 8.14. Suppose that u satisfies the initial-boundary value problem
ut − ∆u = f, in Ω × R+,
∂u
∂n
= g, on Γ × R+,
u(·, 0) = v, in Ω,
where Ω ⊂ Rd is a bounded domain in Rd with a smooth boundary Γ and
∂u/∂n is the exterior normal derivative. Assume in addition that f (x, t) ≥ 0,
v(x) ≥ 0 for x ∈ Ω, t ≥ 0, and g(x, t) > 0 for x ∈ Γ , t ≥ 0. Show that
u(x, t) ≥ 0 for x ∈ Ω, t ≥ 0. (In fact, it is sufficient to assume that g(x, t) ≥ 0.)

8.5 Problems 127
Problem 8.15. Consider the Stokes equations describing the 2-dimensional
motion of a viscous and incompressible fluid at small Reynolds number R:
(8.36)
∂u
∂t
− R−1∆u + ∇p = 0, in R2 × R+,
∇ · u = 0, in R2 × R+,
u(·, 0) = v, in R2,
where u(x, t) ∈ R2 is the dimensionless velocity and p(x, t) ∈ R the dimen-
sionless pressure. In this form R−1 represents the viscosity. Let us define the
vorticity ω by
ω = ∇ × u = ∂u2/∂x1 − ∂u1/∂x2.
Show that (8.36) can be rewritten in the vorticity variable as
∂ω
∂t
− R−1∆ω = 0, in R2 × R+,
ω(·, 0) = ∇ × v, in R2.
Problem 8.16. Let u(x, t) = (E(t)v)(x) be the solution of (8.10) and let
{λj}
∞
j=1 and {ϕj}
∞
j=1 be the eigenvalues and normalized eigenfunctions of
(6.5) as in Theorem 6.4. Show that
u(x, t) = (E(t)v)(x) =
∫
Ω
G(x, y, t)v(y) dy,
where the Green’s function is
G(x, y, t) =
∞∑
j=1
e−λj tϕj (x)ϕj (y).
Hint: see Problem 6.7.

9 Finite Difference Methods for Parabolic
Problems
In this chapter we give an introduction to the numerical solution of parabolic
equations by finite differences, and consider the application of such methods
to the homogeneous heat equation in one space dimension. We first discuss, in
Sect. 9.1, the pure initial value problem, with data given on the unrestricted
real axis, and then in Sect. 9.2 the mixed initial-boundary value problem on a
finite interval in space, under Dirichlet boundary conditions. We discuss sta-
bility and error bounds for various choices of finite difference approximations,
in maximum-norm by maximum principle type arguments and in L2-norm by
Fourier analysis. For the unrestricted problem we consider explicit schemes,
and on a finite interval also implicit ones, such as the Crank-Nicolson scheme.
9.1 The Pure Initial Value Problem
Consider first the pure initial value problem to find u = u(x, t) such that
(9.1)
∂u
∂t
=
∂2u
∂x2
, in R × R+,
u(·, 0) = v, in R,
where v is a given smooth bounded function. We recall from Sect. 8.1 that this
problem has a unique solution, many properties of which may be deduced,
for instance, from the representation
(9.2) u(x, t) =
1
√
4πt
∫ ∞
−∞
e−y
2/4tv(x − y) dy =
(
E(t)v
)
(x),
where E(t) denotes the solution operator of (9.1). In particular, we note
that the solution is bounded with respect to the maximum-norm, or, more
precisely,
(9.3) ∥u(·, t)∥C = ∥E(t)v∥C ≤ ∥v∥C = sup
x∈R
|v(x)|, for t ≥ 0.
For the numerical solution of this problem by finite differences one intro-
duces a grid of mesh-points (x, t) = (xj , tn). Here xj = jh, tn = nk, where j

130 9 Finite Difference Methods for Parabolic Problems
and n are integers, n ≥ 0, h the mesh-width in x, and k the time step, with
both h and k small. One then seeks an approximate solution U nj at these
mesh-points, determined by an equation obtained by replacing the deriva-
tives in (9.1) by difference quotients. For functions defined on the grid we
introduce thus the forward and backward difference quotients with respect
to x,
∂xU
n
j = h
−1(U nj+1 − U
n
j ) and ∂̄xU
n
j = h
−1(U nj − U
n
j−1),
and similarly with respect to t, for instance,
∂tU
n
j = k
−1(U n+1j − U
n
j ).
The simplest finite difference scheme corresponding to (9.1) is then the for-
ward Euler method
∂tU
n
j = ∂x∂̄xU
n
j , for j, n ∈ Z, n ≥ 0,
U 0j = vj := v(xj ), for j ∈ Z,
where Z denotes the integers. The difference equation may also be written
U n+1j − U
n
j
k
=
U nj+1 − 2U
n
j + U
n
j−1
h2
,
or, if we introduce the mesh-ratio λ = k/h2,
(9.4) U n+1j = (EkU
n)j = λU
n
j−1 + (1 − 2λ)U
n
j + λU
n
j+1,
which defines the local discrete solution operator Ek. We shall consider h and
k related by k/h2 = λ = constant, and may therefore omit the dependence
on h in the notation. The scheme (9.4) is called explicit, since it expresses
the solution at t = tn+1 explicitly in terms of the values at t = tn. Iterating
the operator we find that the solution of the discrete problem is
U nj = (E
n
k U
0)j = (E
n
k v)j , for j, n ∈ Z, n ≥ 0.
Assume now that λ ≤ 1
2
. All the coefficients of the operator Ek in (9.4)
are then non-negative, and since their sum is 1, we find
|U n+1j | ≤ λ|U
n
j−1| + (1 − 2λ)|U
n
j | + λ|U
n
j+1| ≤ sup
j∈Z
|U nj |,
so that
sup
j∈Z
|U n+1j | ≤ sup
j∈Z
|U nj |.
Defining for mesh-functions v = (vj ), a discrete maximum-norm by
(9.5) ∥v∥∞,h = sup
j∈Z
|vj|,

9.1 The Pure Initial Value Problem 131
we thus have
∥U n+1∥∞,h = ∥EkU
n
∥∞,h ≤ ∥U
n
∥∞,h,
and hence by repeated application
(9.6) ∥U n∥∞,h = ∥E
n
k v∥∞,h ≤ ∥v∥∞,h,
which is a discrete analogue of the estimate (9.3) for the continuous problem.
The boundedness of the discrete solution operator is referred to as the
stability of this operator. We shall now see that if λ is chosen as a constant
bigger than 1
2
, then the method is unstable. To see this, we choose vj =
(−1)j ϵ, where ϵ is a small positive number, so that ∥v∥∞,h = ϵ. Then
U 1j =
(
λ(−1)j−1 + (1 − 2λ)(−1)j + λ(−1)j+1
)
ϵ = (1 − 4λ)(−1)j ϵ,
or, more generally,
U nj = (1 − 4λ)
n(−1)j ϵ,
whence
∥U n∥∞,h = (4λ − 1)
nϵ → ∞, as n → ∞.
We thus find that in this case, even though the initial data are very small, the
norm of the discrete solution tends to infinity as n → ∞ when k = t/n → 0,
even if t = tn is bounded. This may be interpreted to mean that very small
perturbations of the initial data (for instance by round-off errors) may cause
so big changes in the discrete solution at later times that it becomes useless.
We now restrict the considerations to the stable case, λ ≤ 1
2
, and we
shall show that the discrete solution converges to the exact solution as the
mesh-parameters tend to zero, provided the initial data, and thus the exact
solution of (9.1), are smooth enough. In order to demonstrate this, we need
to use that the exact solution satisfies the difference equation except for
a small error, which tends to zero with h and k. More precisely, setting
unj = u(xj , tn) we have by Taylor’s formula for the solution of (9.1), with
appropriate x̄j ∈ (xj−1, xj+1), t̄n ∈ (tn, tn+1),
τ nj = ∂tu
n
j − ∂x∂̄xu
n
j =
(
∂tu
n
j − ut(xj , tn)
)
−
(
∂x∂̄xu
n
j − uxx(xj , tn)
)
= 1
2
kutt(xj , t̄n) −
1
12
h2uxxxx(x̄j , tn).
Since utt = uxxxx and since one easily sees from (9.2) that |u(·, t)|C4 ≤ |v|C4
for the solution of (9.1), we obtain
∥τ n∥∞,h ≤ Ck max
t∈In
|utt(·, t)|C + Ch
2
|u(·, tn)|C4
≤ Ch2 max
t∈In
|u(·, t)|C4 ≤ Ch
2
|v|C4 , for λ ≤
1
2
.
(9.7)
The expression τ nj is referred to as the truncation error (or local discretization
error ). We now have the following error estimate.

132 9 Finite Difference Methods for Parabolic Problems
Theorem 9.1. Let U n and u be the solutions of (9.4) and (9.1), and let
k/h2 = λ ≤ 1
2
. Then there is constant C such that
∥U n − un∥∞,h ≤ Ctnh
2
|v|C4 for tn ≥ 0.
Proof. Set zn = U n − un. Then
∂tz
n
j − ∂x∂̄xz
n
j = −τ
n
j ,
and hence
zn+1j = (Ekz
n)j − kτ
n
j .
By repeated application this yields
znj = (E
n
k z
0)j − k
n−1∑
l=0
(En−1−lk τ
l)j .
Since z0j = U
0
j − u
0
j = vj − vj = 0 we find, using the stability estimate (9.6)
and the truncation error estimate (9.7),
∥zn∥∞,h ≤ k
n−1∑
l=0
∥τ l∥∞,h ≤ C nk h
2
|v|C4 ,
which is the desired result. ⊓.
The method described has first order accuracy in time and second order
in space, but since k and h are tied by k/h2 = λ ≤ 1
2
, the total effect is
second order accuracy with respect to the mesh-width h.
More generally we may consider finite difference operators of the form
(9.8) U n+1j = (EkU
n)j :=
∑
p
apU
n
j−p, for j, n ∈ Z, n ≥ 0,
where ap = ap(λ), λ = k/h
2, and where the sum is finite. One may associate
with this operator the trigonometric polynomial
(9.9) Ẽ(ξ) =
∑
p
ape
−ipξ.
This polynomial is relevant to the stability analysis and is called the symbol
or the characteristic polynomial of Ek. We find at once the following result.
Theorem 9.2. A necessary condition for stability of the operator Ek in (9.8)
with respect to the discrete maximum-norm defined in (9.5) is that
(9.10) |Ẽ(ξ)| ≤ 1, for ξ ∈ R.

9.1 The Pure Initial Value Problem 133
Proof. Assume that Ek is stable and that |Ẽ(ξ0)| > 1 for some ξ0 ∈ R. Then,
for vj = e
ijξ0 ϵ,
U 1j = ϵ
∑
p
ape
i(j−p)ξ0 = Ẽ(ξ0)vj ,
and by repeated application this yields
∥U n∥∞,h = |Ẽ(ξ0)|
nϵ → ∞, as n → ∞.
Since ∥v∥∞,h = ϵ this contradicts the stability and proves the theorem. ⊓.
For the finite difference operator defined in (9.4) we have Ẽ(ξ) = 1 − 2λ +
2λ cos ξ, and since cos ξ varies in [−1, 1] the condition (9.10) is equivalent to
1 − 4λ ≥ −1, or λ ≤ 1
2
, which agrees with our previous stability condition.
The condition (9.10) is a special case of von Neumann’s stability condition.
We shall see that in a slightly different setting this condition is also sufficient
for stability.
By its definition the symbol of a discrete solution operator is particu-
larly suited for investigating finite difference methods in the framework of
Fourier analysis. It is then convenient to use the l2-norm to measure the
mesh-functions. Let thus V = {Vj}
∞
−∞ be a mesh-function in the space vari-
able and set
∥V ∥2,h =
(
h
∞∑
j=−∞
V 2j
)1/2
.
The set of mesh-functions normed in this way and with finite norm will be
denoted by l2,h. Let us also define for such a mesh-function its discrete Fourier
transform
V̂ (ξ) = h
∞∑
j=−∞
Vj e
−ijξ,
where we assume that the sum is absolutely convergent. The function V̂ (ξ)
is 2π-periodic and V can be retrieved from V̂ (ξ) by
Vj =
1
2πh
∫ π
−π
V̂ (ξ)eijξ dξ.
We recall Parseval’s relation
(9.11) ∥V ∥22,h =
1
2πh
∫ π
−π
|V̂ (ξ)|2 dξ =
1
2π
∫ π/h
−π/h
|V̂ (hξ)|2 dξ.
We may now define stability with respect to the norm ∥ · ∥2,h, or stability
in l2,h, to mean, analogously to (9.6), but allowing a constant factor C on
the right,
(9.12) ∥Enk V ∥2,h ≤ C∥V ∥2,h, for n ≥ 0, h ∈ (0, 1),
and find the following:

134 9 Finite Difference Methods for Parabolic Problems
Theorem 9.3. Von Neumann’s condition (9.10) is a necessary and sufficient
condition for stability of the operator Enk in l2,h.
Proof. We note that
(EkV )̂ (ξ) = h
∑
j
∑
p
apVj−pe
−ijξ
=
∑
p
ape
−ipξh
∑
j
Vj−pe
−i(j−p)ξ = Ẽ(ξ)V̂ (ξ).
Hence
(Enk V )̂ (ξ) = Ẽ(ξ)
nV̂ (ξ),
and using Parseval’s relation (9.11), the stability of Ek in l2,h is equivalent
to ∫ π
−π
|Ẽ(ξ)|2n|V̂ (ξ)|2 dξ ≤ C2
∫ π
−π
|V̂ (ξ)|2 dξ, for n ≥ 0,
for all admissible V̂ . But this is easily seen to hold if and only if
|Ẽ(ξ)|n ≤ C, for n ≥ 0, ξ ∈ R,
which is equivalent to (9.10) (and we thus have C = 1 in (9.12)). ⊓.
In the discussion of an explicit finite difference method of the form (9.8)
it is sometimes convenient to consider the functions of the space variable x to
be defined not just at the mesh-points, but for all x ∈ R, so that we are given
an initial function U 0(x) = v(x) and seek an approximate solution U n(x) at
t = tn, n = 1, 2, . . ., from
(9.13) U n+1(x) = (EkU
n)(x) =
∑
p
apU
n(x − xp), ap = ap(λ), λ = k/h
2.
One advantage of this point of view is that all U n then lie in the same function
space, independently of h, for instance in L2(R) or C(R).
We consider briefly the situation in which the analysis takes place in
L2 = L2(R) and set, allowing now also complex-valued functions,
∥u∥ =
(∫ ∞
−∞
|u(x)|2 dx
)1/2
.
We shall then use the Fourier transform defined by (cf. Appendix A.3)
(9.14) (Fv)(ξ) = v̂(ξ) =
∫ ∞
−∞
v(x)e−ixξ dx,
and note that here, with Ẽ(ξ) defined by (9.9),

9.1 The Pure Initial Value Problem 135
(Ekv)̂ (ξ) =
∑
p
ap(Fv(· − ph))(ξ) =
(∑
p
ape
−iphξ
)
v̂(ξ) = Ẽ(hξ)v̂(ξ).
Recalling Parseval’s relation for (9.14),
∥v∥2 = (2π)−1∥v̂∥2,
we thus find
∥U n∥ = (2π)−1/2∥Ẽ(hξ)nv̂∥ ≤ sup
ξ∈R
|Ẽ(hξ)|n∥v∥,
and therefore that stability with respect to L2 holds if and only if
sup
ξ∈R
|Ẽ(hξ)|n ≤ C, n ≥ 0,
which is again equivalent to von Neumann’s condition (9.10).
Also the convergence analysis may be expressed in L2. We say that the
finite difference operator Ek defined in (9.13) is accurate of order r if
(9.15) Ẽ(ξ) = e−λξ
2
+ O(|ξ|r+2), as ξ → 0.
For instance, for the operator defined in (9.4) we have
Ẽ(ξ) = 1 − 2λ + 2λ cos ξ = 1 − λξ2 + 1
12
λξ4 + O(ξ6)
= e−λξ
2
+
(
1
12
λ − 1
2
λ2
)
ξ4 + O(ξ6),
so that (9.4) is accurate of order 2, or, for the special choice λ = 1
6
, of order
4.
By comparing the coefficients in the Taylor expansion of Ẽ(ξ) − e−λξ
2
around ξ = 0 with those in the expansion of Eku(x, t) − u(x, t + k) around
(x, t), with k = λh2, it is easy to see that the definition (9.15) is equivalent
to saying that, for the exact solution of (9.1),
(9.16) un+1(x) − Eku
n(x) = kO(hr), as h → 0, λ = k/h2 = constant,
i.e., that the one step discrete solution operator approximates the exact so-
lution operator to order kO(hr), see Problem 9.1.
We have then the following result, where we recall that | · |s = | · |Hs .
Theorem 9.4. Assume that Ek is defined by (9.13) with λ = k/h
2 =
constant and is accurate of order r and stable in L2. Then
∥U n − un∥ ≤ Ctnh
r
|v|r+2, for tn ≥ 0.

136 9 Finite Difference Methods for Parabolic Problems
Proof. Since Ẽ(ξ) is bounded on R, we have by (9.15)
|Ẽ(ξ) − e−λξ
2
| ≤ C|ξ|r+2, for ξ ∈ R.
By stability it follows that
(9.17) |Ẽ(ξ)n −e−nλξ
2
| = |(Ẽ(ξ)−e−λξ
2
)
n−1∑
j=0
Ẽ(ξ)n−1−j e−jλξ
2
| ≤ Cn|ξ|r+2.
Now, by Fourier transformation of (9.1) with respect to x, we have as in
Sect. 8.1, that
dû
dt
(ξ, t) = −ξ2û(ξ, t), for t > 0, with û(ξ, 0) = v̂(ξ),
and hence
û(ξ, t) = e−ξ
2tv̂(ξ).
We conclude that
(U n − un)̂ (ξ) =
(
Ẽ(hξ)n − e−nkξ
2)
v̂(ξ),
and hence
∥U n − un∥2 = (2π)−1∥(Ẽ(hξ)n − e−nkξ
2
)v̂(ξ)∥2.
Now, by (9.17)
|Ẽ(hξ)n − e−nkξ
2
| ≤ Cnhr+2|ξ|r+2,
so that, using the facts that (dv/dx)̂ (ξ) = −iξv̂(ξ) and λ = k/h2,
∥U n − un∥ ≤ (2π)−1/2Cnhr+2∥ξr+2v̂(ξ)∥ ≤ C nk hr∥v(r+2)∥.
This shows the conclusion of the theorem under the assumption that the
initial data are such that v(r+2) belongs to L2. In fact, by a more precise
argument, using the smoothing property of the solution operator E(t). one
may reduce this regularity requirement by almost two derivatives. ⊓.
In the above discussion we have only considered finite difference schemes
of one-step (or two-level) type, that is, schemes that use the values at time
t = tn to compute the approximate solution at t = tn+1. It would also
be natural to replace the derivatives in the model heat equation (9.1) by
difference quotients in a symmetric fashion around (x, tn), which would result
in the equation
(9.18)
U n+1(x) − U n−1(x)
2k
= ∂x∂̄xU
n(x).
In this case, in addition to U 0 = v, we also need to prescribe U 1 (presumably
by some approximation of u(·, k)) in order to be able to use (9.18) to find U n

9.1 The Pure Initial Value Problem 137
for n ≥ 0. This two-step, or three-level, scheme would formally be accurate
of second order in both x and t. Although the particular scheme (9.18) turns
out to be unstable for any combination of h and k (cf. Problem 9.6), other
multi-step schemes are useful in applications. For instance, one may show
that the scheme (9.18) may be stabilized, for any constant λ, by replacing
U n(x) on the right by the average 1
2
(U n+1(x)+U n−1(x)) , so that the scheme
becomes the Dufort-Frankel scheme
U n+1(x) − U n−1(x)
2k
=
U n(x + h) − U n+1(x) − U n−1(x) + U n(x − h)
h2
.
We shall end this discussion by making an observation concerning the
accuracy of the Dufort-Frankel scheme. Let thus u be a smooth function and
replace U by u above. With ∂x∂̄x as before and correspondingly for ∂t∂̄t and
with ∂̂t denoting the symmetric difference quotient
∂̂tu(x, t) =
u(x, t + k) − u(x, t − k)
2k
= 1
2
(∂t + ∂̄t)u(x, t),
we have for the truncation error
τh,k,n(x)
=
un+1(x) − un−1(x)
2k
−
un(x + h) − un+1(x) − un−1(x) + un(x − h)
h2
= ∂̂tu(x, tn) − ∂x∂̄xu(x, tn) +
k2
h2
∂t∂̄tu(x, tn)
= (ut − uxx)(x, tn) + O(k
2) + O(h2) +
k2
h2
utt(x, tn) + O
(k4
h2
)
.
Consistency with the heat equation therefore requires that k/h tends to zero,
which is the case, for instance, if k/h2 = λ = constant. However, if instead
k/h = λ = constant, we obtain
τh,k,n(x) = (ut − uxx + λ
2utt)(x, tn) + O(h
2), as h → 0,
which shows that the scheme is then consistent, not with the heat equation,
but with the second order hyperbolic equation
λ2utt + ut − uxx = 0.
Much of the analysis of this section generalizes to the initial-value problem
for the inhomogeneous equation,
ut = uxx + f (x, t), in R × R+,
u(·, 0) = v, in R.
For instance, we may apply the forward Euler scheme

138 9 Finite Difference Methods for Parabolic Problems
∂tU
n
j = ∂x∂̄xU
n
j + f
n
j , for j, n ∈ Z, n ≥ 0,
U 0j = vj := v(xj ), for j ∈ Z,
or, with Ek defined as in (9.4),
U n+1j = (EkU
n)j + k f
n
j .
One may conclude at once by the stability of Ek in maximum-norm that
∥U n∥∞,h ≤ ∥v∥∞,h + k
n−1∑
l=0
∥f l∥∞,h.
Moreover, in the same way as in the proof of Theorem 9.1 one may easily
prove the error estimate
∥U n − un∥∞,h ≤ Ctnh
2 max
s≤tn
(
|utt(·, s)|C + |u(·, s)|C4
)
.
9.2 The Mixed Initial-Boundary Value Problem
In many physical situations our above pure initial value model problem (9.1)
is inadequate, and instead it is required to solve the heat equation on a finite
interval with boundary values given at the end points of the interval for
positive time. We thus have reason to consider the following model problem
(9.19)
ut = uxx, in Ω = (0, 1), t > 0,
u(0, t) = u(1, t) = 0, for t > 0,
u(·, 0) = v, in Ω.
For the approximate solution we may again cover the domain with a grid
of mesh-points, this time by dividing Ω into subintervals of equal length
h = 1/M , where M is a positive integer, and setting (xj , tn) = (jh, nk)
with j = 0, . . . , M and n = 0, 1, . . . . With U nj denoting the approximation of
u(xj , tn), the explicit forward Euler scheme is now
(9.20)
∂tU
n
j = ∂x∂̄xU
n
j , for j = 1, . . . , M − 1, n ≥ 0,
U n0 = U
n
M = 0, for n > 0,
U 0j = Vj = v(xj ), for j = 0, . . . , M,
or, for U nj , j = 0, . . . , M , given,
U n+1j = λ(U
n
j−1 + U
n
j+1) + (1 − 2λ)U
n
j , j = 1, . . . , M − 1,
U n+10 = U
n+1
M = 0.

9.2 The Mixed Initial-Boundary Value Problem 139
In this case we are thus looking for a sequence of (M + 1)-vectors U n =
(U n0 , . . . , U
n
M )
T with U n0 = U
n
M = 0 satisfying these equations. In the analysis
we shall first use the discrete maximum-norm
∥U n∥∞,h = max
0≤j≤M
|U nj |.
When λ = k/h2 ≤ 1
2
we conclude, as for the pure initial value problem, that
∥U n+1∥∞,h ≤ ∥U
n
∥∞,h,
or, defining the local solution operator Ek in the obvious way,
∥Enk V ∥∞,h ≤ ∥V ∥∞,h, for n ≥ 0.
The scheme is thus maximum-norm stable for λ ≤ 1
2
.
In order to see that this condition is necessary for stability also in the
present case, we modify our counter-example from Sect. 9.1 so as to incorpo-
rate the boundary conditions and set
U 0j = Vj = (−1)
j sin(πjh), for j = 0, . . . , M.
By a simple calculation analogous to that of the proof of Theorem 9.2 we
then have
U nj = (1 − 2λ − 2λ cos(πh))
nVj , for j = 0, . . . , M.
If λ > 1
2
we have for h sufficiently small
|1 − 2λ − 2λ cos(πh)| ≥ γ > 1,
and hence, if tn = 1, say,
∥U n∥∞,h ≥ γ
n
∥V ∥∞,h → ∞, as h → 0.
In the presence of stability we may derive an error estimate in the same
way as for the pure initial value problem. The estimate in (9.7) now shows
for the truncation error τ nj = ∂tu
n
j − ∂x∂̄xu
n
j ,
|τ nj | ≤ Ch
2 max
t∈In
|u(·, t)|C4 , where In = (tn, tn+1),
and we obtain the following error estimate.
Theorem 9.5. Let U n and u be the solutions of (9.20), with λ ≤ 1
2
, and
(9.19). Then
∥U n − un∥∞,h ≤ Ctnh
2 max
t≤tn
|u(·, t)|C4 , for tn ≥ 0.

140 9 Finite Difference Methods for Parabolic Problems
We remark that in this case, in order for u to be sufficiently regular to
guarantee that the right hand side of (9.7) is bounded by Ch2|v|C4 , we need to
require certain compatibility conditions for v with the boundary conditions,
namely v(x) = v′′(x) = v(iv)(x) = 0 for x = 0, 1.
We note that a method of the form
U n+1j =
∑
p
apU
n
j−p, for j = 1, . . . , M − 1,
is not suitable here if ap ̸= 0 for some |p| > 1, since then for some interior
mesh-point of Ω the equation uses mesh-points outside this interval. In such
a case the equation has to be modified near the endpoints, which significantly
complicates the analysis.
The stability requirement k ≤ 1
2
h2 used for the forward Euler method is
quite restrictive in practice, and it would be desirable to relax it to be able
to use h and k of the same order of magnitude. For this purpose one may
define an implicit method, instead of the explicit method considered above,
by the backward Euler scheme
(9.21)
∂̄tU
n+1
j = ∂x∂̄xU
n+1
j , for j = 1, . . . , M − 1, n ≥ 0,
U n+10 = U
n+1
M = 0, for n ≥ 0,
U 0j = Vj = v(xj ), for j = 0, . . . , M.
For U n given this may be put in the form
(1 + 2λ)U n+1j − λ(U
n+1
j−1 + U
n+1
j+1 ) = U
n
j , j = 1, . . . , M − 1,
U n+10 = U
n+1
M = 0,
which is a linear system of equations for the determination of U n+1. In matrix
notation it may be written as
(9.22) BŪ n+1 = Ū n,
where Ū n+1 and Ū n are now thought of as vectors with M − 1 components
corresponding to the interior mesh-points and B is the diagonally dominant,
symmetric, tridiagonal matrix
B =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 + 2λ −λ 0 . . . 0
−λ 1 + 2λ −λ
. . .
…
0
. . .
. . .
. . . 0
…
. . . −λ 1 + 2λ −λ
0 . . . 0 −λ 1 + 2λ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Clearly the system (9.22) may easily be solved for Ū n+1.

9.2 The Mixed Initial-Boundary Value Problem 141
Introducing the finite dimensional space l0h of (M + 1)-vectors {Vj}
M
j=0
with V0 = VM = 0, and the operator Bkh on l
0
h defined by
(BkhV )j = (1 + 2λ)Vj − λ(Vj−1 + Vj+1) = Vj − k∂x∂̄xVj , j = 1, . . . , M − 1,
we may write the above system may as
BkhU
n+1 = U n,
or, again with Ek denoting the local solution operator,
U n+1 = B−1kh U
n = EkU
n.
We shall now show that this method is stable in maximum-norm without any
restrictions on k and h, or, more precisely,
(9.23) ∥U n+1∥∞,h ≤ ∥U
n
∥∞,h, for n ≥ 0.
In fact, with suitable j0,
∥U n+1∥∞,h = |U
n+1
j0
| ≤
1
1 + 2λ
(
λ
(
|U n+1j0−1| + |U
n+1
j0+1
|
)
+ |U nj0|
)
≤
2λ
1 + 2λ
∥U n+1∥∞,h +
1
1 + 2λ
∥U n∥∞,h,
from which (9.23) follows at once. This implies the stability estimate
(9.24) ∥U n∥∞,h = ∥E
n
k V ∥∞,h ≤ ∥V ∥∞,h.
The solution operator Enk is thus stable in maximum-norm and convergence
of U n to u(tn) may also be proved. This time we have for the truncation error
τ nj = ∂̄tu
n+1
j − ∂x∂̄xu
n+1
j = O(k + h
2), as k, h → 0, for j = 1, . . . , M − 1,
where, since h and k are unrelated, the latter expression does not reduce to
O(h2). As a consequence the convergence result now reads as follows.
Theorem 9.6. Let U n and u be the solutions of (9.19) and (9.21). Then
∥U n − un∥∞,h ≤ C tn(h
2 + k) max
t≤tn
|u(·, t)|C4 , for tn ≥ 0.
Proof. Defining the error zn = U n − un we may write
Bkhz
n+1 = BkhU
n+1
− Bkhu
n+1 = U n − (un+1 − k∂x∂̄xu
n+1) = zn − k τ n,
where we consider τ n to be an element of l0h. Thus
zn+1 = Ekz
n
− kEkτ
n,

142 9 Finite Difference Methods for Parabolic Problems
and hence
zn = −k
n−1∑
l=0
En−lk τ
l.
The estimate (9.7) is now replaced by
∥τ n∥∞,h ≤ C(h
2 + k) max
t∈In−1
|u(·, t)|C4 .
Using (9.24) we then obtain
∥zn∥∞,h ≤ k
n−1∑
l=0
∥τ l∥∞,h ≤ C tn(h
2 + k) max
t≤tn
|u(·, t)|C4 ,
which concludes the proof. ⊓.
The above convergence result for the backward Euler method is satisfac-
tory in that it requires no restriction on the mesh-ratio λ = k/h2. On the
other hand, since it is only first order accurate in time, the error in the time
discretization will dominate unless k is chosen much smaller than h. It would
thus be desirable to find a stable method which is second order accurate
also with respect to time. Such a method is provided by the Crank-Nicolson
scheme, which was introduced for a system of ordinary differential equations
in Sect. 7.2. This uses symmetry around the point (xj , tn+1/2) and is defined
by
(9.25)
∂̄tU
n+1
j =
1
2
∂x∂̄x
(
U nj + U
n+1
j
)
, for j = 1, . . . , M − 1, n ≥ 0,
U n+10 = U
n+1
M = 0, for n ≥ 0,
U 0j = Vj := v(jh), for j = 0, . . . , M.
The first equation may also be written
(I − 1
2
k∂x∂̄x)U
n+1
j = (I +
1
2
k∂x∂̄x)U
n
j ,
or
(1 + λ)U n+1j −
1
2
λ(U n+1j−1 + U
n+1
j+1 ) = (1 − λ)U
n
j +
1
2
λ(U nj−1 + U
n
j+1),
and, in matrix form, with Ū n again denoting the (M − 1)-vector associated
with U n,
BŪ n+1 = AŪ n,
where now both A and B are symmetric tridiagonal matrices, with B diago-
nally dominant:
B =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 + λ − 1
2
λ 0 . . . 0
−
1
2
λ 1 + λ − 1
2
λ
. . .
…
0
. . .
. . .
. . . 0
…
. . . − 1
2
λ 1 + λ − 1
2
λ
0 . . . 0 − 1
2
λ 1 + λ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,

9.2 The Mixed Initial-Boundary Value Problem 143
and
A =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 − λ 1
2
λ 0 . . . 0
1
2
λ 1 − λ 1
2
λ
. . .
…
0
. . .
. . .
. . . 0
…
. . . 1
2
λ 1 − λ 1
2
λ
0 . . . 0 1
2
λ 1 − λ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
With obvious notation we also have
BkhU
n+1 = AkhU
n,
or
U n+1 = B−1kh AkhU
n = EkU
n,
where, similarly to the above,
∥B−1kh V ∥∞,h ≤ ∥V ∥∞,h.
The same approach to stability as for the backward Euler method gives,
for λ ≤ 1, since the coefficients on the right are then non-negative,
(1 + λ)∥U n+1∥∞,h ≤ λ∥U
n+1
∥∞,h + ∥U
n
∥∞,h,
or
∥U n+1∥∞,h ≤ ∥U
n
∥∞,h,
which shows stability. However, if λ > 1, which is the interesting case if we
want to be able to take h and k of the same order, one obtains instead
(1 + λ)∥U n+1∥∞,h ≤ λ∥U
n+1
∥∞,h + (2λ − 1)∥U
n
∥∞,h,
which does not yield maximum-norm stability, since 2λ − 1 > 1. For λ ≤ 1
we have immediately as before a O(k2 + h2) = O(h2) convergence estimate.
In order to be able to deal with λ > 1, we now instead turn to an analysis
in an l2 type norm. We introduce thus for vectors V = (V0, . . . , VM )
T the
inner product
(V, W )h = h
M∑
j=0
Vj Wj ,
and the corresponding norm
∥V ∥2,h = (V, V )
1/2
h =
(
h
M∑
j=0
V 2j
)1/2
.
We denote by l02,h the space l
0
h equipped with this inner product and norm,
and note that this space is spanned by the M −1 vectors ϕp, p = 1, . . . , M −1,
with components

144 9 Finite Difference Methods for Parabolic Problems
ϕp,j =
√
2 sin(πpjh), for j = 0, . . . , M,
and that these form an orthonormal basis with respect to the above inner
product (cf. Problem 9.7), i.e.,
(ϕp, ϕq)h = δpq =
{
1, if p = q,
0, if p ̸= q.
We also observe that the ϕp are eigenfunctions of the finite difference opera-
tors −∂x∂̄x ,
−∂x∂̄xϕp,j =
2
h2
(1 − cos(πph))ϕp,j , for j = 1, . . . , M − 1.
We shall now to discuss the stability within this framework of the three
difference schemes considered above. Let V be given initial data in l02,h. Then
V =
M−1∑
p=1
V̂pϕp, where V̂p = (V, ϕp)h.
The forward Euler method then gives
U 1j = Vj + k∂x∂̄xVj =
M−1∑
p=1
V̂p
(
1 − 2λ(1 − cos(πph))
)
ϕp,j , j = 1, . . . , M − 1,
with U 10 = U
1
M = 0, or, more generally,
(9.26) U nj =
M−1∑
p=1
V̂pẼ(πph)
nϕp,j , j = 0, . . . , M,
where Ẽ(ξ) is the symbol of the local discrete solution operator Ek,
Ẽ(ξ) = 1 − 2λ + 2λ cos ξ.
By Parseval’s relation we have thus
∥U n∥2,h =
( M−1∑
p=1
V̂ 2p Ẽ(πph)
2n
)1/2
≤ max
p
|Ẽ(πph)n| ∥V ∥2,h,
with equality for the appropriate V . Now for 1 ≤ p ≤ M − 1 we have
|Ẽ(πph)| = max
{
|1 − 2λ(1 − cos(πh))|, |1 − 2λ(1 − cos(π(M − 1)h))|
}
= max
{
|1 − 2λ(1 − cos(πh))|, |1 − 2λ(1 + cos(πh))|}.
We thus have maxp |Ẽ(phπ)| ≤ 1 for small h if and only if 4λ − 1 ≤ 1, or
λ ≤ 1
2
, and it follows in this case that

9.2 The Mixed Initial-Boundary Value Problem 145
(9.27) ∥U n∥2,h ≤ ∥V ∥2,h.
Consequently, the forward Euler scheme is stable in l02,h if and only if λ ≤
1
2
,
i.e., under the same conditions as for the maximum-norm.
The corresponding analysis for the backward Euler scheme gives (9.26),
where now
Ẽ(ξ) =
1
1 + 2λ(1 − cos ξ)
.
In this case 0 ≤ Ẽ(πph) ≤ 1 for all p and λ and hence (9.27) is valid for any
value of λ.
Similarly, for the Crank-Nicolson scheme, (9.26) holds with
Ẽ(ξ) =
1 − λ(1 − cos ξ)
1 + λ(1 − cos ξ)
,
and we now note that |Ẽ(ξ)| ≤ 1 and all ξ for any λ > 0. Thus, the Fourier
analysis method shows stability in l02,h for any λ. The convergence follows
again by the standard method and gives the following.
Theorem 9.7. Let U n and u be the solutions of (9.25) and (9.19). Then
∥U n − un∥2,h ≤ C tn(h
2 + k2) max
t≤tn
|u(·, t)|C6 , for tn ≥ 0,
Proof. We write the truncation error
τ nj = ∂̄tu
n+1
j − ∂x∂̄x
unj + u
n+1
j
2
=
(
∂̄tu
n+1
j − ut(xj , tn+1/2)
)
+ ∂x∂̄x
(unj + u
n+1
j
2
− u
n+1/2
j
)
+
(
∂x∂̄xu
n+1/2
j − uxx(xj , tn+1/2)
)
,
and hence, using Taylor expansions as earlier,
∥τ n∥2,h ≤ C(h
2 + k2) max
t∈In
|u(·, t)|C6 .
In the same way as before the error znj = U
n
j − u
n
j satisfies
zn+1 = U n+1 − un+1 = EkU
n
− B−1kh Bkhu
n+1 = Ekz
n
− k B−1kh τ
n,
or
zn = −k
n−1∑
l=0
En−1−lk B
−1
kh τ
l,
from which the result follows by using the stability of the Crank-Nicolson
operator Enk and the boundedness of B
−1
kh . ⊓.

146 9 Finite Difference Methods for Parabolic Problems
The forward and backward Euler methods and the Crank-Nicolson method
may all be considered to be special cases of the θ-method defined by
(9.28) ∂̄tU
n+1
j = θ∂x∂̄xU
n+1
j + (1 − θ)∂x∂̄xU
n
j , j = 1, . . . , M − 1,
with θ = 0 for the forward Euler, θ = 1 for the backward Euler, and θ = 1/2
for the Crank-Nicolson method. The equation (9.28) may be written as
(I − θk∂x∂̄x)U
n+1 = (I + (1 − θ)k∂x∂̄x)U
n,
and we find this time for the symbol
Ẽ(ξ) =
1 − 2(1 − θ)λ(1 − cos ξ)
1 + 2θλ(1 − cos ξ)
.
Assuming 0 ≤ θ ≤ 1 we have Ẽ(ξ) ≤ 1 for ξ ∈ R, and the stability require-
ment reduces to
min
ξ
Ẽ(ξ) =
1 − 4(1 − θ)λ
1 + 4θλ
≥ −1,
or
(1 − 2θ)λ ≤ 1
2
.
Hence the θ method is unconditionally stable in l02,h, i.e., stable in l
0
2,h for all
λ, if θ ≥ 1/2, whereas for θ < 1/2 stability holds if and only if λ ≤ 1 2(1 − 2θ) . 9.3 Problems Problem 9.1. Show the equivalence of definitions (9.15) and (9.16) of accu- racy of order r. Use the alternate definition (9.16) to show that the accuracy of (9.4) is of order 4, if λ = 1/6. Problem 9.2. Formulate and prove an analogue of Theorem 9.2 in two space dimensions. Problem 9.3. Let (ajk) be a symmetric, positive definite 2 × 2 matrix. For the solution of the initial value problem ∂u ∂t = 2∑ j,k=1 ajk ∂2u ∂xj ∂xk , in R2 × R+, u(·, 0) = v, in R2, we wish to apply the finite difference method 9.3 Problems 147 ∂tU n ij = 2∑ k,l=1 akl ∂xk ∂̄xl U n ij . (a) Give sufficient conditions on the coefficients for the method to be stable in the maximum-norm. (b) Is the method stable in l2,h? Problem 9.4. Find an explicit 5-point finite difference operator for (9.1) of the form (9.8) (i.e., with five terms on the right-hand side of (9.8)) of order of accuracy 4. Discuss the stability of this operator. Problem 9.5. Formulate a finite difference method for ut = ∆u, in R 2 × R+, u(·, 0) = v, in R2, such that ∥U n − un∥∞,h = O(h 4), as h → 0. Problem 9.6. Consider the three-level finite difference method (9.18), and let U 0 = V and U 1 = W , with V, W ∈ L2(R). Show that Û n(ξ) = c1(ξ)τ1(ξ) n + c2(ξ)τ2(ξ) n, where τ1,2(ξ) are the roots of the equation τ 2 + 4λ(1 − cos ξ)τ − 1 = 0, with λ = k/h2, and c1(ξ) and c2(ξ) are determined from c1(ξ) + c2(ξ) = V̂ (ξ), c1(ξ)τ1(ξ) + c2(ξ)τ2(ξ) = Ŵ (ξ). Use this to show that ∥U n∥ → ∞ as n → ∞ for any λ > 0, and thus that
(9.18) is unstable.
Problem 9.7. Let {ϕp}
M−1
p=1 be defined by (9.2). Show that they form an
orthonormal basis for l02,h and that they are eigenfunctions of the differ-
ence operator −∂x∂̄x with eigenvalues 2h
−2(1−cos(πph)). Compare with the
eigenfunctions and eigenvalues of −d2/dx2. Note that one of the ϕp gives the
counter-example to stability in the beginning of Sect. 9.2.
Problem 9.8. (A discrete maximum principle.) Let Ω ⊂ R be a bounded
interval and I = (0, T ]. Show that if λ = kh−2 ≤ 1
2
and
∂tU
n
j − ∂x∂̄xU
n
j ≤ 0, for (xj , tn) ∈ Ω × I,
then U nj attains its maximum on the parabolic boundary Γp, cf. Theorem
8.6. Hint: Use the argument leading to (9.6). Prove a similar result for the
backward Euler method.

148 9 Finite Difference Methods for Parabolic Problems
Problem 9.9. We know that all norms on the finite dimensional space l0h are
equivalent. For example, show that
∥V ∥2,h ≤ ∥V ∥∞,h ≤ h
−1/2
∥V ∥2,h, for V ∈ l
0
h,
and that these inequalities are sharp. Note that the equivalence is not uniform
in h and is lost when h → 0, that is, when the dimension of l0h tends to infinity.
The second inequality above has the same character as the inverse inequality
(6.37), relating a stronger norm (∥ · ∥∞,h) to a weaker norm (∥ · ∥2,h).
Problem 9.10. Show that the function ϕ(x) = eixξ is an eigenfunction of
the differential and difference operators ∂/∂x, ∂x, and ∂̄x.
Problem 9.11. (Computer exercise.) Consider the initial-boundary value
problem (9.19) with v(x) = sin(πx) − sin(3πx). Apply the forward Euler
method with h = 1/10 and k = 1/600, 1/300, 1/100. Apply also the Crank-
Nicolson method with h = k = 1/10. Calculate the error at (1/2, 1).

10 The Finite Element Method for a Parabolic
Problem
In this chapter we consider the approximation of solutions of the model heat
equation in two space dimensions by means of Galerkin’s method, using piece-
wise linear trial functions. In Sect. 10.1 we consider the discretization with
respect to the space variables only, and in the following Sect. 10.2 we study
some completely discrete schemes.
10.1 The Semidiscrete Galerkin Finite Element Method
Let Ω ⊂ R2 be a bounded convex domain with smooth boundary Γ , and
consider the initial-boundary value problem,
(10.1)
ut − ∆u = f, in Ω × R+,
u = 0, on Γ × R+,
u(·, 0) = v, in Ω,
where ut denotes ∂u/∂t and ∆ the Laplacian ∂
2/∂x21 + ∂
2/∂x22. In the first
step we shall approximate the solution u(x, t) by means of a function uh(x, t)
which, for each fixed t, is a piecewise linear function of x over a triangulation
Th of Ω, thus depending on a finite number of parameters.
Thus, let Th = {K} denote a triangulation of Ω of the type considered in
Sect. 5.2 and let {Pj}
Mh
j=1 be the interior nodes of Th. Further, let Sh denote
the continuous piecewise linear functions on Th which vanish on ∂Ω and let
{Φj}
Mh
j=1 be the standard basis of Sh corresponding to the nodes {Pj}
Mh
j=1.
Recall the definition (5.28) of the interpolant Ih : C0(Ω̄) → Sh, and the error
bounds (5.34) with r = 2.
For the purpose of defining thus an approximate solution to the initial
boundary value problem (10.1) we first write this in weak form as in Sect. 8.3,
i.e., with the definitions there,
(10.2) (ut, ϕ) + a(u, ϕ) = (f, ϕ), ∀ϕ ∈ H
1
0 , t > 0.
We then pose the approximate problem to find uh(t) = uh(·, t), belonging to
Sh for each t, such that

150 10 The Finite Element Method for a Parabolic Problem
(10.3)
(uh,t, χ) + a(uh, χ) = (f, χ), ∀χ ∈ Sh, t > 0,
uh(0) = vh,
where vh ∈ Sh is some approximation of v. Since we have discretized only
in the space variables, this is referred to as a spatially semidiscrete problem.
In the next section, we shall discretize also in the time variable to produce
completely discrete schemes.
In terms of the basis {Φj}
Mh
j=1 our semidiscrete problem may be stated:
Find the coefficients αj (t) in
uh(x, t) =
Mh∑
j=1
αj (t)Φj (x),
such that
Mh∑
j=1
α′j (t)(Φj , Φk) +
Mh∑
j=1
αj (t)a(Φj , Φk) = (f (t), Φk), k = 1, . . . , Mh,
and, with γj denoting the nodal values of the given initial approximation vh,
αj (0) = γj , j = 1, . . . , Mh.
In matrix notation this may be expressed as
(10.4) Bα′(t) + Aα(t) = b(t), for t > 0, with α(0) = γ,
where B = (bkj ) is the mass matrix with elements bkj = (Φj , Φk), A = (akj )
the stiffness matrix with akj = a(Φj , Φk), b = (bk) the vector with entries
bk = (f, Φk), α(t) the vector of unknowns αj (t), and γ = (γj ). The dimension
of all these items equals Mh, the number of interior nodes of Th.
We recall from Sect. 5.2 that the stiffness matrix A is symmetric positive
definite, and this holds also for the mass matrix B since
Mh∑
k,j=1
ξj ξk(Φj , Φk) =
∥
∥
∥
Mh∑
j=1
ξj Φj
∥
∥
∥
2
≥ 0,
and since equality can only occur if the vector ξ = 0. In particular, B is
invertible, and therefore the above system of ordinary differential equations
may be written
α′(t) + B−1Aα(t) = B−1b(t), for t > 0, with α(0) = γ,
and hence obviously has a unique solution for t positive.
We begin our analysis by considering the stability of the semidiscrete
method. Since uh(t) ∈ Sh we may choose χ = uh(t) in (10.3) to obtain

10.1 The Semidiscrete Galerkin Finite Element Method 151
(uh,t, uh) + a(uh, uh) = (f, uh), for t > 0,
or, since the first term equals 1
2
d
dt
∥uh∥
2 and the second is non-negative,
1
2
d
dt
∥uh∥
2 = ∥uh∥
d
dt
∥uh∥ ≤ ∥f∥ ∥uh∥.
This yields
d
dt
∥uh∥ ≤ ∥f∥,
which after integration shows the stability estimate
(10.5) ∥uh(t)∥ ≤ ∥vh∥ +
∫ t
0
∥f∥ ds.
For the purpose of writing equation in (10.3) in operator form, we intro-
duce a discrete Laplacian ∆h, which we think of as an operator from Sh into
itself, defined by
(10.6) (−∆hψ, χ) = a(ψ, χ), ∀ψ, χ ∈ Sh.
This discrete analogue of Green’s formula clearly defines ∆hψ =
∑Mh
j=1 dj Φj
from
Mh∑
j=1
dj (Φj , Φk) = −a(ψ, Φk), k = 1, . . . , Mh,
since the matrix of this system is the positive definite mass matrix encoun-
tered above. The operator ∆h is easily seen to be selfadjoint and −∆h is
positive definite in Sh with respect to the L2-inner product, see Problem
10.3. With Ph denoting the L2-projection onto Sh, the equation in (10.3)
may now be written
(uh,t − ∆huh − Phf, χ) = 0, ∀χ ∈ Sh,
or, noting that the first factor is in Sh, so that χ may be chosen equal to it,
it follows that
(10.7) uh,t − ∆huh = Phf, for t > 0, with uh(0) = vh,
We denote by Eh(t) the solution operator of the homogeneous case of
the semidiscrete equation in (10.7), with f = 0. Hence Eh(t) is the operator
which takes the initial data uh(0) = vh into the solution uh(t) at time t, so
that uh(t) = Eh(t)vh. It is then easy to show (cf. Duhamel’s principle (8.22))
that the solution of the initial value problem (10.7) is
(10.8) uh(t) = Eh(t)vh +
∫ t
0
Eh(t − s)Phf (s) ds.

152 10 The Finite Element Method for a Parabolic Problem
We now note that it follows from (10.5) that Eh(t) is stable in L2, or
(10.9) ∥Eh(t)vh∥ ≤ ∥vh∥, ∀vh ∈ Sh.
Since also Ph has unit norm in L2 this, together with (10.8), re-establishes
the stability estimate (10.5) for the inhomogeneous equation, so that, in fact,
it suffices to show stability for the homogeneous equation.
We shall prove the following estimate for the error between the solutions
of the semidiscrete and continuous problems.
Theorem 10.1. Let uh and u be the solutions of (10.3) and (10.1). Then
∥uh(t) − u(t)∥ ≤ ∥vh − v∥ + Ch
2
(
∥v∥2 +
∫ t
0
∥ut∥2 ds
)
, for t ≥ 0.
Here we require, as usual, that the solution of the continuous problem has
the regularity implicitly assumed by the presence of the norms on the right.
Note also that for vh = Ihv, (5.31) shows that
(10.10) ∥vh − v∥ ≤ Ch
2
∥v∥2,
in which case the first term on the right is dominated by the second. The same
holds true if vh = Phv, where Ph denotes the orthogonal projection of L2 onto
Sh, since this choice is the best approximation of v in Sh with respect to the
L2-norm, see (5.39). Another choice of optimal order is vh = Rhv, where Rh
is the elliptic (or Ritz) projection onto Sh defined in (5.49) by
(10.11) a(Rhv, χ) = a(v, χ), ∀χ ∈ Sh.
Thus Rhv is the finite element approximation of the solution of the elliptic
problem whose exact solution is v. We recall the error estimates of Theo-
rem 5.5,
(10.12) ∥Rhv − v∥ + h|Rhv − v|1 ≤ Ch
s
∥v∥s, for s = 1, 2.
We now turn to the
Proof of Theorem 10.1. In the main step of the proof we shall compare the
solution of the semidiscrete problem to the elliptic projection of the exact
solution. We write
(10.13) uh − u = (uh − Rhu) + (Rhu − u) = θ + ρ.
The second term is easily bounded using (10.12) and obvious estimates by
∥ρ(t)∥ ≤ Ch2∥u(t)∥2 = Ch
2
∥
∥
∥v +
∫ t
0
ut ds
∥
∥
∥
2
≤ Ch2
(
∥v∥2 +
∫ t
0
∥ut∥2 ds
)
.
In order to bound θ, we note that

10.1 The Semidiscrete Galerkin Finite Element Method 153
(θt, χ) + a(θ, χ) = (uh,t, χ) + a(uh, χ) − (Rhut, χ) − a(Rhu, χ)
= (f, χ) − (Rhut, χ) − a(u, χ) = (ut − Rhut, χ),
(10.14)
or
(10.15) (θt, χ) + a(θ, χ) = −(ρt, χ), ∀χ ∈ Sh.
In this derivation we have used (10.3), (10.2), the definition of Rh in (10.11),
and the easily established fact that this operator commutes with time dif-
ferentiation, i.e., Rhut = (Rhu)t. We may now apply the stability estimate
(10.5) to (10.15) to obtain
∥θ(t)∥ ≤ ∥θ(0)∥ +
∫ t
0
∥ρt∥ ds.
Here
∥θ(0)∥ = ∥vh − Rhv∥ ≤ ∥vh − v∥ + ∥Rhv − v∥ ≤ ∥vh − v∥ + Ch
2
∥v∥2,
and further
∥ρt∥ = ∥Rhut − ut∥ ≤ Ch
2
∥ut∥2.
Together these estimates prove the theorem. ⊓.
We see from the proof of Theorem 10.1 that the error estimate for the
semidiscrete parabolic problem is thus a consequence of the stability for this
problem combined with the error estimate for the elliptic problem, expressed
in terms of ρ = (Rh − I)u .
Recalling the maximum principle for parabolic equations, Theorem 8.7,
we find at once that, for the solution operator E(t) of the homogeneous case of
the initial boundary value problem (10.1), we have ∥E(t)v∥C ≤ ∥v∥C for t ≥ 0.
The corresponding maximum principle does not hold for the finite element
problem, but it may be shown that, if the family {Th} of triangulations is
quasi-uniform, cf. (5.52), then for some C > 1,
∥Eh(t)vh∥C ≤ C∥vh∥C, for t ≥ 0.
This may be combined with the error estimate (5.53) for the stationary prob-
lem to show a maximum-norm error estimate for the parabolic problem.
In this regard we mention a variant of the semidiscrete problem (10.2)
for which a maximum principle sometimes holds, namely the lumped mass
method. To define this we replace the matrix B in (10.4) by a diagonal matrix
B̄, in which the diagonal elements are the row sums of B. One can show that
this method can also be defined by
(10.16) (uh,t, χ)h + a(uh, χ) = (f, χ), ∀χ ∈ Sh, for t > 0,
where the inner product in the first term has been obtained by computing
the first term in (10.2) by using the nodal quadrature rule (5.64). For this

154 10 The Finite Element Method for a Parabolic Problem
method one may derive a O(h2) error estimate similar to that of Theorem
10.1. If we now assume that all angles of the triangulations are ≤ π/2, then
the off-diagonal elements of the stiffness matrix A are nonpositive, and as a
result of this one may show that, if Ēh(t) denotes the solution operator of
the modified problem, then
∥Ēh(t)vh∥C ≤ ∥vh∥C, for t ≥ 0.
This is a discrete maximum principle, which is not true for the standard finite
element method.
Returning to the standard Galerkin method (10.3) we now prove the
following estimate for the error in the gradient.
Theorem 10.2. Under the assumptions of Theorem 10.1, we have for t ≥ 0,
|uh(t) − u(t)|1 ≤ |vh − v|1 + Ch
{
∥v∥2 + ∥u(t)∥2 +
(∫ t
0
∥ut∥
2
1 ds
)1/2}
.
Proof. As before we write the error in the form (10.13). Here by (10.12),
|ρ(t)|1 = |Rhu(t) − u(t)|1 ≤ Ch∥u(t)∥2.
In order to estimate ∇θ we use again (10.15), now with χ = θt. We obtain
∥θt∥
2 + 1
2
d
dt
|θ|21 = −(ρt, θt) ≤
1
2
(∥ρt∥
2 + ∥θt∥
2),
so that
d
dt
|θ|21 ≤ ∥ρt∥
2,
or
|θ(t)|21 ≤ |θ(0)|
2
1 +
∫ t
0
∥ρt∥
2 ds ≤
(
|vh − v|1 + |Rhv − v|1
)2
+
∫ t
0
∥ρt∥
2 ds.
Hence, since a2 + b2 ≤ (|a| + |b|)2 and in view of (10.12), we conclude
(10.17) |θ(t)|1 ≤ |vh − v|1 + Ch
{
∥v∥2 +
(∫ t
0
∥ut∥
2
1 ds
)1/2}
,
which completes the proof. ⊓.
Note that if vh = Ihv or Rhv, then
|vh − v|1 ≤ Ch∥v∥2,
so that the first term on the right in Theorem 10.2 is dominated by the
second.

10.1 The Semidiscrete Galerkin Finite Element Method 155
We make the following observation concerning θ = uh − Rhu: Assume
that we choose vh = Rhv, so that θ(0) = 0. Then in addition to (10.17) we
have
|θ(t)|1 ≤
(∫ t
0
∥ρt∥2 ds
)1/2
≤ Ch2
(∫ t
0
∥ut∥
2
2 ds
)1/2
.
Hence the gradient of θ is of second order O(h2), whereas the gradient of the
total error is only of order O(h) as h → 0. Thus ∇uh is a better approximation
to ∇Rhu than is possible to ∇u. This is an example of a phenomenon which
is sometimes referred to as superconvergence.
The discrete solution operator Eh(t) introduced above also has smooth-
ing properties analogous to the corresponding results in Sect. 8.2 for the
continuous problem, such as, for instance
|Eh(t)vh|1 ≤ Ct
−1/2
∥vh∥, for t > 0, vh ∈ Sh,
and
(10.18)
∥
∥
∥Dkt Eh(t)vh
∥
∥
∥ = ∥∆khEh(t)vh∥ ≤ Ckt
−k
∥vh∥, for t > 0, vh ∈ Sh.
Such results may be used to show, e.g., the following non-smooth data error
estimate for the homogeneous equation.
Theorem 10.3. Assume that f = 0 and let uh and u be the solutions of
(10.3) and (10.1), respectively, where now the initial data for (10.3) are cho-
sen as vh = Phv. Then
∥uh(t) − u(t)∥ ≤ Ch
2t−1∥v∥, for t > 0.
The proof is left as an exercise (Problem 10.4). This result shows that the
convergence rate is O(h2) for t bounded away from zero, even when v is only
assumed to belong to L2.
The above theory easily extends to finite elements of higher order, under
the appropriate regularity assumptions on the solution. Thus, if the finite
element subspace is such that
(10.19) ∥Rhw − w∥ ≤ Ch
r
∥w∥r, ∀w ∈ H
r
∩ H10 ,
then we may show the following theorem.
Theorem 10.4. Let uh and u be the solutions of (10.3) and (10.1), respec-
tively, and assume that (10.19) holds. Then, for vh suitably chosen, we have
∥uh(t) − u(t)∥ ≤ Ch
r
(
∥v∥r +
∫ t
0
∥ut∥r ds
)
, for t ≥ 0.
Recall from (5.50) that for r > 2 the estimate (10.19) holds for piecewise
polynomials of degree r − 1, but that the regularity assumption w ∈ Hr ∩ H10
is then somewhat unrealistic for a polygonal domain Ω. For a domain Ω with
a smooth boundary Γ , special considerations are needed in the boundary
layer Ω \ Ωh.

156 10 The Finite Element Method for a Parabolic Problem
10.2 Some Completely Discrete Schemes
We shall now turn our attention to some simple schemes for discretization
also with respect to the time variable, and let Sh be the space of piecewise
linear finite element functions as before. We begin with the backward Euler-
Galerkin method. With k the time step and U n ∈ Sh the approximation of
u(t) at t = tn = nk, this method is defined by replacing the time derivative
in (10.3) by a backward difference quotient, or with ∂̄tU
n = k−1(U n −U n−1),
(10.20)
(∂̄tU
n, χ) + a(U n, χ) = (f (tn), χ), ∀χ ∈ Sh, n ≥ 1,
U 0 = vh.
Given U n−1 this defines U n implicitly from the discrete elliptic problem
(U n, χ) + ka(U n, χ) = (U n−1 + kf (tn), χ), ∀χ ∈ Sh.
Expressing U n in terms of the basis {Φj}
Mh
j=1 as U
n(x) =
∑Mh
j=1 α
n
j Φj (x), we
may write this equation in the matrix notation introduced in Sect. 10.1 as
Bαn + kAαn = Bαn−1 + kbn, for n ≥ 1,
where αn is the vector with components αnj , or
αn = (B + kA)−1Bαn−1 + k(B + kA)−1bn, for n ≥ 1, with α0 = γ.
We begin our analysis of the backward Euler method by showing that it
is unconditionally stable, i.e., that it is stable independently of the relation
between h and k. Choosing χ = U n in (10.20) we have, since a(U n, U n) ≥ 0,
(∂̄tU
n, U n) ≤ ∥f n∥ ∥U n∥, where f n = f (tn),
or
∥U n∥2 − (U n−1, U n) ≤ k∥f n∥ ∥U n∥.
Since (U n−1, U n) ≤ ∥U n−1∥ ∥U n∥, this shows
∥U n∥ ≤ ∥U n−1∥ + k∥f n∥, for n ≥ 1,
and hence, by repeated application,
(10.21) ∥U n∥ ≤ ∥U 0∥ + k
n∑
j=1
∥f j∥.
We shall now prove the following error estimate.
Theorem 10.5. With U n and u the solutions of (10.20) and (10.1), respec-
tively, and with vh chosen so that (10.10) holds, we have, for n ≥ 0,
∥U n − u(tn)∥ ≤ Ch
2
(
∥v∥2 +
∫ tn
0
∥ut∥2 ds
)
+ Ck
∫ tn
0
∥utt∥ ds.

10.2 Some Completely Discrete Schemes 157
Proof. In analogy with (10.13) we write
U n − u(tn) =
(
U n − Rhu(tn)
)
+
(
Rhu(tn) − u(tn)
)
= θn + ρn.
As before, by (10.12),
∥ρn∥ ≤ Ch2∥u(tn)∥2 ≤ Ch
2
(
∥v∥2 +
∫ tn
0
∥ut∥2 ds
)
.
This time, a calculation corresponding to (10.14) yields
(10.22) (∂̄tθ
n, χ) + a(θn, χ) = −(ωn, χ),
where
ωn = Rh∂̄tu(tn) − ut(tn) = (Rh − I)∂̄tu(tn) + (∂̄tu(tn) − ut(tn)) = ω
n
1 + ω
n
2 .
By application of the stability estimate (10.21) to (10.22) we obtain
∥θn∥ ≤ ∥θ0∥ + k
n∑
j=1
∥ω
j
1∥ + k
n∑
j=1
∥ω
j
2∥.
Here, as before, by (10.10) and (10.12),
∥θ0∥ = ∥vh − Rhv∥ ≤ ∥vh − v∥ + ∥v − Rhv∥ ≤ Ch
2
∥v∥2.
Note now that
ω
j
1 = (Rh − I)k
−1
∫ tj
tj−1
ut ds = k
−1
∫ tj
tj−1
(Rh − I)ut ds,
whence
k
n∑
j=1
∥ω
j
1∥ ≤
n∑
j=1
∫ tj
tj−1
Ch2∥ut∥2 ds = Ch
2
∫ tn
0
∥ut∥2 ds.
Further, by Taylor’s formula,
ω
j
2 = k
−1(u(tj ) − u(tj−1)) − ut(tj ) = −k
−1
∫ tj
tj−1
(s − tj−1)utt(s) ds,
so that
k
n∑
j=1
∥ω
j
2∥ ≤
n∑
j=1
∥
∥
∥
∫ tj
tj−1
(s − tj−1)utt(s) ds
∥
∥
∥ ≤ k
∫ tn
0
∥utt∥ ds.
Together our estimates complete the proof of the theorem. ⊓.

158 10 The Finite Element Method for a Parabolic Problem
Replacing the backward difference quotient with respect to time in (10.20)
by a forward difference quotient we arrive at the forward Euler-Galerkin
method, or with ∂tU
n = (U n+1 − U n)/k,
(∂tU
n, χ) + a(U n, χ) = (f (tn), χ), ∀χ ∈ Sh, n ≥ 1,
U 0 = vh.
In matrix form this may be expressed as
Bαn+1 = (B − kA)αn + kbn, for n ≥ 0,
Since B is not a diagonal matrix this method is not explicit. However, if
this time discretization method is applied to the lumped mass semidiscrete
equation (10.16), and thus B replaced by the diagonal matrix B̄, then the
corresponding forward Euler method becomes an explicit one.
Using the discrete Laplacian defined in (10.6), the forward Euler method
may also be defined by
(10.23) U n+1 = (I + k∆h)U
n + kPhf (tn), for n ≥ 0, with U
0 = vh.
This method is not unconditionally stable as the backward Euler method,
but considering for simplicity only the homogeneous equation, we shall show
stability under the condition that the family {Sh} is such that
(10.24) λMh,h k ≤ 2,
where λMh,h is the largest eigenvalue of −∆h. Recalling (6.38), we note that
this holds, e.g., if the Sh satisfy the inverse inequality (6.37) and if k ≤
2C−1h2, where C is the constant in (6.38), which thus shows conditional
stability.
It is clear that (10.23) is stable if and only if ∥(I + k∆h)χ∥ ≤ ∥χ∥ for
all χ ∈ Sh, and since −∆h is symmetric positive definite, this holds if and
only if all eigenvalues of I + k∆h belong to [−1, 1]. By the positivity of −∆h
this is the same as requiring the smallest eigenvalue of I + k∆h to be ≥ −1,
or that the largest eigenvalue of −∆h is ≤ 2/k, which is (10.24). See also
Problem 10.3.
Note that because of the non-symmetric choice of the discretization in
time, the backward Euler-Galerkin method is only first order accurate in time.
We therefore now turn to the Crank-Nicolson-Galerkin method, in which the
semidiscrete equation is discretized in a symmetric fashion around the point
tn−1/2 = (n −
1
2
)k, which yields a method which is second order accurate in
time. More precisely, we define U n ∈ Sh recursively for n ≥ 1 by
(10.25)
(∂̄tU
n, χ) + a( 1
2
(U n + U n−1), χ) = (f (tn−1/2), χ), ∀χ ∈ Sh,
U 0 = vh.

10.3 Problems 159
In matrix notation this takes the form,
Bαn + 1
2
kAαn = Bαn−1 − 1
2
kAαn−1 + kbn−1/2, for n ≥ 1,
or, with α0 = γ,
αn = (B + 1
2
kA)−1(B − 1
2
kA)αn−1 + k(B + 1
2
kA)−1bn−1/2, n ≥ 1.
This method is also unconditionally stable which may be shown by choos-
ing χ = U n + U n−1 in (10.25) and using the Cauchy-Schwarz inequality on
the right. Then
k(∂̄tU
n, U n+U n+1) = ∥U n∥2−∥U n−1∥2 = (∥U n∥−∥U n−1∥)(∥U n∥+∥U n−1∥).
Using the positivity of a(U n, U n) and cancelling ∥U n∥ + ∥U n−1∥ we find
∥U n∥ ≤ ∥U n−1∥ + k∥f n−1/2∥, where f n−1/2 = f (tn−1/2),
or after summation
∥U n∥ ≤ ∥vh∥ + k
n∑
j=1
∥f j−1/2∥.
This time the error estimate reads as follows. Its proof is similar to that
of Theorem 10.5 and is left to Problem 10.7.
Theorem 10.6. With U n and u the solutions of (10.25) and (10.1), respec-
tively, and with vh chosen so that (10.10) holds, we have for n ≥ 0,
∥U n − u(tn)∥ ≤ Ch
2
(
∥v∥2 +
∫ tn
0
∥ut∥2 ds
)
+ Ck2
∫ tn
0
(
∥uttt∥ + ∥∆utt∥
)
ds.
10.3 Problems
Problem 10.1. Consider the problem (10.1) in the case of one space dimen-
sion with Ω = (0, 1). For the numerical solution, we use the piecewise linear
functions based on the partition
0 < x1 < x2 < . . . < xM < 1, xj = jh, h = 1/(M + 1). Determine the mass matrix B and the stiffness matrix A and write down the semidiscrete problem, the backward Euler equations, and the Crank-Nicolson equations. Problem 10.2. (Computer exercise.) Consider the initial boundary value problem (10.1) with Ω = (−π, π) and v = sign x. 160 10 The Finite Element Method for a Parabolic Problem (a) Determine the exact solution by eigenfunction expansion. (b) Apply the backward Euler method (10.20) based on piecewise linear finite elements with vh = Phv and (h, k) = (π/5, 1/10), (π/10, 1/40). Determine the maximal error at the mesh-points for t = 0.1, 0.5, 1.0. Problem 10.3. (a) Show that the operator −∆h : Sh → Sh defined in (10.6) is selfadjoint positive definite with respect to (·, ·). (b) Show that, with the notation of Theorem 6.7, −∆hvh = Mh∑ i=1 λi,h(vh, ϕi,h)ϕi,h and ∥∆h∥ = λMh,h. Hint: The left side of the second identity is the operator norm of ∆h, see (A.7). Thus, you must show that ∥∆hχ∥ ≤ λMh,h∥χ∥ for all χ ∈ Sh with equality for some χ. (c) Assume that the family of finite element spaces {Sh} satisfies the inverse inequality (6.37). Show that ∥∆h∥ ≤ Ch −2. Hint: See (6.38). Problem 10.4. Assume that f = 0 and let uh and u be the solutions of (10.3) and (10.1), respectively, with vh = Phv. (a) Assume that v ∈ H2 ∩ H10 . Show that ∥uh(t) − u(t)∥ ≤ Ch 2 ∥v∥2, for t ≥ 0. (b) Assume that v ∈ L2. Show that ∥uh(t) − u(t)∥ ≤ Ch 2t−1∥v∥, for t > 0.
Hint: For (a) deduce from (10.15) that
(10.26) θ(t) = Eh(t)θ(0) −
∫ t
0
Eh(t − s)Phρt(s) ds.
Split the integral as
∫ t
0
=
∫ t/2
0
+
∫ t
t/2
and integrate by parts in the first term
to get, with e = uh − u,
θ(t) = Eh(t)Phe(0) − Eh(t/2)Phρ(t/2)
+
∫ t/2
0
DsEh(t − s)Phρ(s) ds −
∫ t
t/2
Eh(t − s)Phρt(s) ds.
Then use (10.18), (10.12), (8.18), and Problem 8.10. Note also Phe(0) = 0,
since vh = Phv.

10.3 Problems 161
For (b) integrate by parts once more to get the additional terms
DtEh(t/2)Phρ̃(t/2) −
∫ t/2
0
D2s Eh(t − s)Phρ̃(s) ds,
where ρ̃(t) =
∫ t
0
ρ(s) ds, ∥ρ̃∥ ≤ Ch2∥ũ∥2, ∥ũ∥2 ≤ C∥∆ũ∥, and ∆ũ(t) =∫ t
0
ut(s) ds = u(t) − v.
Problem 10.5. Assume that the family of finite element spaces {Sh} is such
that ∥∆h∥ ≤ Ch
−2, cf. Problem 10.3. Let uh and u be the solutions of (10.3)
and (10.1), respectively. Assume that ∥vh − v∥ ≤ Ch
2∥v∥2. Show that
∥uh(t) − u(t)∥ ≤ C
(
1 + log(t/h2)
)
h2 max
0≤s≤t
∥u(s)∥2, for t ≥ h
2.
Hint: Integrate by parts in (10.26) to get
θ(t) = Eh(t)Phe(0) − Phρ(t) +
∫ t
0
DsEh(t − s)Phρ(s) ds.
Split the integral as
∫ t
0
=
∫ t−h2
0
+
∫ t
t−h2 and treat the first part as in Prob-
lem 10.4 (a). For the second part use ∥DsEh(t − s)∥ = ∥∆hEh(t − s)∥ ≤
∥∆h∥∥Eh(t − s)∥ ≤ Ch
−2, see Problem 10.3 (b).
Problem 10.6. Show error estimates analogous to those of Theorem 10.1
when the term −∆u in (10.1) is replaced by Au = −∇ · (a∇u) + b · ∇u + cu
as in Sect. 3.5. Hint: See Problems 5.7 and 8.8.
Problem 10.7. Prove Theorem 10.6.

11 Hyperbolic Equations
In this chapter we present basic concepts and results for hyperbolic equa-
tions. We begin in Sect. 11.1 with a short discussion of characteristic direc-
tions, curves, and surfaces. In Sect. 11.2 we study the model wave equation.
We use the method of eigenfunction expansions to solve the standard initial
boundary value problem, and apply the energy method to study uniqueness
and domains of dependence. In Sect. 11.3 we reduce the solution of first order
scalar first order partial differential equations to integration along character-
istic curves, and in Sect. 11.4 we extend this approach to symmetric first order
system, and consider finally symmetric hyperbolic systems in more than one
space variable by energy arguments.
11.1 Characteristic Directions and Surfaces
Consider the scalar linear partial differential equation
(11.1) Lu = L(x, D)u :=
∑
|α|≤m
aα(x)D
αu = f (x), in Ω,
where Ω is a domain in Rd. We say that the direction ξ ∈ Rd, ξ ̸= 0, is a
characteristic direction for the operator L(x, D) at x if
(11.2) Λ(ξ) = Λ(x, ξ) :=
∑
|α|=m
aα(x)ξ
α = 0.
The polynomial Λ(ξ) = Λ(x, ξ) is called the characteristic polynomial of L at
x. Note that the summation in (11.2) is only over |α| = m, i.e., it corresponds
to the principal part of L, the terms of order exactly m.
Sometimes we shall consider also systems of linear partial differential
equations. These may be included in (11.1) if we interpret the coefficients
aα(x) as matrices. In the case that these matrices are square matrices of
order N with N ≥ 2, we say that ξ ∈ Rd is a characteristic direction at x if
(11.3) det Λ(x, ξ) = 0.
A (d − 1)-dimensional surface in Rd is said to be a characteristic surface
if its normal at each point x is a characteristic direction at x. In the case of
the plane, d = 2, we call this a characteristic curve or simply a characteristic.

164 11 Hyperbolic Equations
Example 11.1. For the first order scalar equation
(11.4)
d∑
j=1
aj (x)
∂u
∂xj
+ a0(x)u = f (x),
the characteristic directions are given by the equation
Λ(x, ξ) =
d∑
j=1
aj (x)ξj = 0,
and hence any direction orthogonal to the vector a(x) = (a1(x), . . . , ad(x)) is
characteristic.
The hyperplane x1 = 0 has the normal (1, 0, . . . , 0) and hence it is a
characteristic surface if a1(x) = 0 for all x = (0, x2, . . . , xd). It is non-
characteristic if a1(x) ̸= 0 at each point x = (0, x2, . . . , xd), which is equiva-
lent to saying that the equation (11.4) may be solved for ∂u/∂x1. In such a
case the equation may be written
∂u
∂x1
=
d∑
j=2
ãj (x)
∂u
∂xj
+ ã0(x)u + f̃ (x)
near the hyperplane.
Example 11.2. Poisson’s equation,
−∆u = f,
has no characteristic directions, since Λ(ξ) = −(ξ21 +· · ·+ξ
2
d) = −|ξ|
2 vanishes
only for ξ = 0.
Example 11.3. The heat equation,
∂u
∂t
− ∆u = f,
now considered in Rd+1 with points (x, t), x ∈ Rd, t ∈ R, has the character-
istic equation Λ(ξ, τ ) = −|ξ|2 = 0. In this case, the variable is (ξ, τ ) ∈ Rd+1,
ξ ∈ Rd, τ ∈ R, which means that (0, . . . , 0, 1) is a characteristic direction
and the hyperplane t = 0 a characteristic surface.
Example 11.4. The wave equation,
∂2u
∂t2
− ∆u = f,
similarly corresponds to Λ(ξ, τ ) = τ 2 − |ξ|2 = 0, so that (ξ, ±|ξ|) is a charac-
teristic direction for any choice of ξ ̸= 0. For instance, the circular cone with
vertex (x̄, t̄), defined by the equation

11.1 Characteristic Directions and Surfaces 165
F (x, t) := |x − x̄|2 − (t − t̄)2 = 0,
has for its normal at a point (x, t) on the cone
( ∂F
∂x1
, . . . ,
∂F
∂xd
,
∂F
∂t
)
= 2(x − x̄, −(t − t̄)) = 2(x − x̄, ∓|x − x̄|).
This is thus a characteristic direction and the cone itself a characteristic
surface. Now t = 0 is non-characteristic.
The characteristic polynomial may be used to classify partial differen-
tial equations into different types. For example, L is said to be elliptic if it
has no characteristic directions. For a second order equation with constant
coefficients, Λ(ξ) is a homogeneous quadratic polynomial, so that
Λ(ξ) =
d∑
j,k=1
ajkξj ξk, where ajk = akj .
After an orthogonal transformation of variables, ξ = P η, such a polynomial
may be written in the form
Λ(P η) =
d∑
j=1
λj η
2
j ,
where {λj}
d
j=1 are the eigenvalues of the matrix A = (ajk). The differen-
tial equation is said to be elliptic if all the λj are of the same sign, as in
Example 11.2, which is equivalent to the above definition that it has no char-
acteristics. The equation is said to be hyperbolic, if all but one of the λj have
the same sign and the remaining λj has the opposite sign, as in Example 11.4.
In Example 11.3, all but one λj have the same sign and the remaining λj is
zero, and we then have a parabolic equation.
Example 11.5. Let now A be an N × N diagonal matrix with diagonal ele-
ments {λj}
N
j=1, and consider the system
∂u
∂t
− A
∂u
∂x
= f.
The characteristic directions (ξ, τ ) are then determined by the equation
det (τ I − ξA) = 0.
The matrix τ I −ξA is a diagonal matrix with elements τ −λj ξ, j = 1, . . . , N ,
and thus (ξ, τ ) is a characteristic direction exactly when one of these elements
vanishes. This gives the characteristic directions (1, λj ), j = 1, . . . , N . Thus
the straight lines
x + λj t = constant, j = 1, . . . , N,
are characteristic curves and t = 0 is non-characteristic.

166 11 Hyperbolic Equations
11.2 The Wave Equation
In this section we first consider the initial-boundary value problem for the
wave equation,
(11.5)
utt − ∆u = 0, in Ω × R+,
u = 0, on Γ × R+,
u(·, 0) = v, ut(·, 0) = w, in Ω,
where Ω is a bounded domain in Rd with boundary Γ , and v and w are given
functions of x in Ω.
The existence of a solution of (11.5) may be shown by eigenfunction ex-
pansion, in a way analogous to the case of the heat equation in Sect. 8.2. For
the purpose of demonstrating this, we introduce the eigenfunctions {ϕj}
∞
j=1
and corresponding eigenvalues {λj}
∞
j=1 of the elliptic operator −∆ and as-
sume that (11.5) has a solution the form
u(x, t) =
∞∑
j=1
ûj (t)ϕj (x).
Inserting this into the differential equation we find
∞∑
j=1
(
û′′j (t) + λj ûj (t)
)
ϕj (x) = 0.
Correspondingly, we have for the initial conditions
∞∑
j=1
ûj (0)ϕj (x) = v(x),
∞∑
j=1
û′j (0)ϕj (x) = w(x).
Since the ϕj form an orthonormal basis of L2 = L2(Ω) we have, for j ≥ 1,
û′′j + λj ûj = 0, for t > 0,
ûj (0) = v̂j = (v, ϕj ), û
′
j (0) = ŵj = (w, ϕj ),
and by solving this initial-value problem we conclude
ûj (t) = v̂j cos(
√
λj t) + ŵj
1
√
λj
sin(
√
λj t), for j ≥ 1,
and hence
(11.6) u(x, t) =
∞∑
j=1
(
v̂j cos(
√
λj t) + ŵj λ
−1/2
j sin(
√
λj t)
)
ϕj (x).
It is clear that if v and w are sufficiently regular for the series to converge
also after differentiation, then this represents a solution of (11.5), see Prob-
lem 11.4. We have thus arrived at the following.

11.2 The Wave Equation 167
Theorem 11.1. Assume that v ∈ H2 ∩ H10 , w ∈ H
1
0 . Then the series (11.6)
is a solution of (11.5).
We shall now prove an energy estimate for the solution u of (11.5). By this
estimate we easily obtain, in the standard way, the uniqueness and stability
of the solution of the problem. The energy method described here is useful
also in situations when the eigenfunction expansion approach does not apply.
Theorem 11.2. Let u = u(x, t) be a sufficiently smooth solution of (11.5).
Then the total energy E(t) of u is constant in time, i.e.,
(11.7) E(t) := 1
2
∫
Ω
(
u2t + |∇u|
2
)
dx = E(0).
Proof. Multiplying the differential equation in (11.5) by ut and integrating
with respect to x over Ω, using also Green’s formula, we find
∫
Ω
uttut dx +
∫
Ω
∇u · ∇ut dx = 0,
or, with the notation of Sect. 8.3,
(utt, ut) + a(u, ut) = 0,
Hence,
1
2
d
dt
∥ut∥
2 + 1
2
d
dt
∥∇u∥2 = 0,
or
d
dt
E(t) = 0, for t > 0.
This immediately implies the statement of the theorem. ⊓.
We shall now prove an energy estimate for the pure initial value problem
for the wave equation, from which we infer that the solution at a given point
(x, t) with t > 0 only depends on the initial data in a certain sphere in the
initial plane t = 0. The problem considered is then
(11.8)
utt − ∆u = 0, in R
d
× R+,
u(·, 0) = v, ut(·, 0) = w, in R
d.
Theorem 11.3. Let u be a solution of the wave equation in (11.8). For (x̄, t̄)
a given point in Rd × R+, let K denote the circular cone, cf. Fig. 11.1,
(11.9) K =
{
(x, t) ∈ Rd × R+ : |x − x̄| ≤ t̄ − t, t ≤ t̄
}
,
and set
EK (t) =
1
2
∫
Bt
(
ut(x, t)
2 + |∇u(x, t)|2
)
dx,
where Bt =
{
x ∈ Rd : (x, t) ∈ K
}
. Then
EK (t) ≤ EK (0), for 0 ≤ t ≤ t̄.

168 11 Hyperbolic Equations
x1
x2
B0
Bt
Kt
t
(x̄, t̄)
Fig. 11.1. The light cone.
Proof. We introduce the mantle surface of K, M =
{
(x, t) : |x − x̄| = t̄ − t
}
,
and set Mt =
{
(x, τ ) ∈ M : τ ≤ t
}
. By multiplication of the differential
equation by 2ut we find
0 = 2
(
utt − ∇ · ∇u
)
ut = 2uttut + 2∇u · ∇ut − 2∇ ·
(
∇uut
)
= Dt
(
u2t + |∇u|
2
)
− 2∇ ·
(
∇uut
)
.
Integrating over Kt =
{
(x, τ ) ∈ K : 0 ≤ τ ≤ t
}
, cf. Fig. 11.1, and using
the divergence theorem, we obtain, with n = (nx, nt) = (nx1 , . . . , nxd , nt) the
exterior normal of ∂Kt,
0 =
∫
∂Kt
(
nt
(
u2t + |∇u|
2
)
− 2nx · ∇u ut
)
ds
=
∫
Bt
(
u2t + |∇u|
2
)
dx −
∫
B0
(
u2t + |∇u|
2
)
dx
+
∫
Mt
(
nt
(
u2t + |∇u|
2
)
− 2ut nx · ∇u
)
ds.
To complete the proof we now show that the integrand of the last term is
nonnegative. We have n2t = |nx|
2 on M , and because nt = 1/
√
2 this yields,
by the Cauchy-Schwarz inequality
|nx · ∇u| ≤ |nx| |∇u| = nt |∇u|.
Using also the inequality 2 ab ≤ a2 + b2, we obtain
2 |ut nx · ∇u| = 2 |ut| |nx · ∇u| ≤ 2 nt |ut| |∇u| ≤ nt
(
u2t + |∇u|
2
)
,
which completes the proof. ⊓.

11.3 First Order Scalar Equations 169
It follows from Theorem 11.3 that, if v = w = 0 in B0, then u = 0 in K,
and thus, in particular, at (x̄, t̄). This shows that the value of the solution of
(11.8) at (x̄, t̄) depends only on the values of v and w in the ball B0 defined
by the circular cone K with vertex (x̄, t̄), and not on the values of v and w
outside this ball.
The existence of a solution of the pure initial value problem (11.8) may be
shown in different ways. For the particular equation considered here, an ex-
plicit solution may be written down in the form of an integral representation,
which takes different forms depending on the number d of space dimensions.
For instance, for d = 1, it is easy to verify, cf. Problem 11.5, that
(11.10) u(x, t) = 1
2
(
v(x + t) + v(x − t)
)
+ 1
2
∫ x+t
x−t
w(y) dy,
which is called d’Alembert’s formula, and for d = 3 one may show
u(x, t) =
∂
∂t
{ 1
4πt
∫
|y−x|=t
v(y) dsy
}
+
1
4πt
∫
|y−x|=t
w(y) dsy.
In this case the solution at (x, t) depends only on the values of the data on
the sphere cut out by the characteristic cone at time zero. More generally,
this holds when d is an odd integer. When d is even the solution at (x, t)
depends on the initial data in the “ball” |y − x| ≤ t.
11.3 First Order Scalar Equations
We now turn to the first order scalar differential equation
(11.11)
d∑
j=1
aj (x)
∂u
∂xj
+ a0(x)u = f (x), x ∈ Ω,
where Ω ⊂ Rd is a bounded domain with boundary Γ , the vector field a =
a(x) = (a1(x), . . . , ad(x)) is smooth and does not vanish at any point, and
a0, f are given smooth functions.
We say that x = x(s) = (x1(s), . . . , xd(s)), with s a real parameter, is a
characteristic curve, or simply a characteristic, for (11.11) if
(11.12)
d
ds
x(s) = a(x(s)),
that is, if the curve in Rd defined by x = x(s) has the vector a(x) as a tangent
at each of its points. Note that a characteristic direction is a normal to the
characteristic curve. In particular, in the special case d = 2, a characteristic
is a characteristic curve in the sense described in Sect. 11.1.

170 11 Hyperbolic Equations
Γ0
Γ+
Γ
−
Fig. 11.2. Inflow and outflow boundaries.
In coordinate form (11.12) may be written as the system of ordinary
differential equations
dxj
ds
= aj (x), for j = 1, . . . , d,
and, since the vector field does not vanish, it is clear from the theory of such
equations that for each x0 ∈ Ω there exists a unique such curve in some
neighborhood of x0 such that x(0) = x0.
Let Γ be the boundary of Ω and denote by Γ− the inflow boundary defined
by
Γ− =
{
x ∈ Γ : n(x) · a(x) < 0 } , where n(x) is the exterior normal to Γ at x. Through each point of Γ− there is a unique characteristic which enters Ω, and we prescribe for the solution of (11.11) the boundary condition (11.13) u = v, on Γ−, where v is a given smooth function on Γ−. We also introduce the outflow and the characteristic boundaries, Γ+ = { x ∈ Γ : n(x) · a(x) > 0
}
, Γ0 =
{
x ∈ Γ : n(x) · a(x) = 0
}
.
Consider now a solution u of (11.11), (11.13) along a characteristic x =
x(s), i.e., consider the function w(s) = u(x(s)). We have by the chain rule

11.3 First Order Scalar Equations 171
dw
ds
= ∇u ·
dx
ds
= a(x) · ∇u,
so that, by (11.11), w satisfies
(11.14)
dw
ds
+ a0(x(s))w = f (x(s)), for s > 0,
w(0) = v(x0), with x(0) = x0 ∈ Γ−.
This is an initial value problem for a linear ordinary differential equation,
which may be solved for the value of w at the points along the characteristic.
To find the solution of (11.11), (11.13) at a point x̄ ∈ Ω we thus determine
the characteristic through x̄, find its intersection x0 with Γ−, and then solve
the equation (11.14) with x(0) = x0. The solution at x̄ thus only depends on
v(x0) and the values of f on the characteristic.
In the special case that a0 = f = 0 in Ω, the equation (11.14) reduces to
dw
ds
= 0, for s > 0, with w(0) = v(x0), x(0) = x0 ∈ Γ−.
Thus in this case u(x(s)) is constant along the characteristic and the value
of the solution at x̄ is the same as at x(0), i.e., u(x(s)) = u(x(0)) = v(x(0)).
This procedure is often referred to as the method of characteristics.
Equations of the form (11.11) are often obtained in the limit from the sta-
tionary heat or diffusion equation with convection when the heat conduction
or diffusion coefficient vanishes, see (1.18). Such equations can be written
in the form (11.11) also in the time-dependent case, if one of the indepen-
dent variables is interpreted as time. Writing the time variable explicitly in
(11.11), we have
ut + a · ∇u + a0u = f, in Ω × R+,
u = g, in Γ−,x,
u(·, 0) = v, in Ω.
Now Ω ⊂ Rd denotes a spatial domain with boundary Γ , and the inflow
boundary of Ω × R+ is split into its spatial part Γ−,x =
{
(x, t) ∈ Γ × R+ :
a(x, t) · n < 0 } and its temporal part Γ−,t = Ω × {0} corresponding to t = 0. We may then use the time variable to parametrize the characteristic curves, x = x(t), which are often called streamlines in this situation. Example 11.6. Consider the problem ut + λux = 0, in R × R+, u(·, 0) = v, in R. Here the characteristics (x(s), t(s)) are determined by 172 11 Hyperbolic Equations dx ds = λ, dt ds = 1. We may thus take t as the parameter along the characteristic and obtain x = λt + C. The characteristic through (x̄, t̄) is (11.15) x − x̄ = λ(t − t̄), and, since the solution is constant on this line, u(x̄, t̄) = v(x̄ − λt̄). Example 11.7. With Ω = (0, 1), we now ask for a solution of ut + λux + u = 1, in Ω × R+, u = 0, on Γ−, where λ = constant > 0. Here
Γ− =
(
{0} × R+
)
∪
(
Ω̄ × {0}
)
= Γ−,x ∪ Γ−,t,
and the characteristic through (x̄, t̄) is again defined by (11.15).
We consider first the case x̄ ≥ λt̄ (see Fig. 11.3). Then the characteristic
through (x̄, t̄) starts at (x̄ − λt̄, 0) ∈ Γ−,x. With s = t as a parameter we
introduce w(s) = u(x̄ + λ(s − t̄), s) and find that the equation for w is
w′ + w = 1, for s > 0, with w(0) = 0.
Hence
(11.16) w(s) = 1 − e−s,
and
u(x̄, t̄) = 1 − e−t̄.
In the case x̄ < λt̄ the characteristic through (x̄, t̄) starts at (0, t̄ − x̄/λ) ∈ Γ−,t and with s = t−(t̄−x̄/λ) as a parameter we find again (11.16) and thus u(x̄, t̄) = 1 − e−x̄/λ. Altogether, we thus have u(x, t) = { 1 − e−t, if x ≥ λt, 1 − e−x/λ, if x < λt. Note that the solution is continuous at x = λt, but that the derivatives ut and ux are not. 11.4 Symmetric Hyperbolic Systems 173 x x < λt x > λt
1
1
t
Fig. 11.3. Example 11.7.
Example 11.8. With the same domain as in Example 11.7 we now consider
the problem
ut + (1 + t)ux = 0, in Ω × R+,
u = x2, on Γ−.
The characteristic through (x̄, t̄) is now (see Fig. 11.4)
x = t + 1
2
t2 + x̄ − t̄ − 1
2
t̄2,
and starts at (x̄ − t̄ − 1
2
t̄2, 0) ∈ Γ−,x, if x̄ ≥ t̄ +
1
2
t̄2, and somewhere on Γ−,t,
if x̄ < t̄ + 1 2 t̄2. The solution is therefore u(x, t) = { (x − t − 1 2 t2)2, if x ≥ t + 1 2 t2, 0, if x < t + 1 2 t2. 11.4 Symmetric Hyperbolic Systems We first consider an initial value problem in one space dimension of the form 174 11 Hyperbolic Equations t x x < t + 1 2 t2 x > t + 1
2
t2
1
1
Fig. 11.4. Example 11.8.
(11.17)
∂u
∂t
+ A(x, t)
∂u
∂x
+ B(x, t)u = f (x, t), for x ∈ R, t > 0,
u(x, 0) = v(x), for x ∈ R,
where u = u(x, t) and f = f (x, t) are N -vector valued functions and A =
A(x, t) and B = B(x, t) are smooth N × N matrices, with A symmetric.
The matrix A then has real eigenvalues {λj}
N
j=1, with λj = λj (x, t), and we
make the additional assumption that these are distinct. The system (11.17) is
then called strictly hyperbolic. Under this assumption one may find a smooth
orthogonal matrix P = P (x, t), which diagonalizes A, so that
P TAP = Λ = diag(λj )
N
j=1,
see Problem 11.17. Introducing a new dependent variable w by u = P w we
find
∂u
∂t
+ A
∂u
∂x
+ Bu = P
∂w
∂t
+ AP
∂w
∂x
+
(∂P
∂t
+ A
∂P
∂x
+ BP
)
w = f,
or,
∂w
∂t
+ Λ
∂w
∂x
+ B̃w = P Tf, where B̃ = P T
(∂P
∂t
+ A
∂P
∂x
+ BP
)
,

11.4 Symmetric Hyperbolic Systems 175
t
x
(x̄, t̄)
x = xN (t) x = xN−1(t) x = x2(t) x = x1(t)
(xN (0), 0) (xN−1(0), 0) (x2(0), 0) (x1(0), 0)
Fig. 11.5. Characteristic curves. Domain of dependence.
which is a system of the form (11.17), but with A diagonal.
We now suppose, thus without restricting the generality, that A in (11.17)
is itself a diagonal matrix, and that the λj are arranged in increasing order,
λ1 < λ2 < · · · < λN . Consider first the case that B = 0. The system then consists of N uncou- pled equations ∂uj ∂t + λj (x, t) ∂uj ∂x = fj (x, t), with uj (x, 0) = vj (x), for j = 1, . . . , N, each of which is a scalar problem of the kind considered in Sect. 11.3 above. Corresponding to each j there exists a characteristic through (x̄, t̄) deter- mined by dx dt = λj (x, t), with x(t̄ ) = x̄. Denoting the solution of this initial value problem by xj (t), so that the char- acteristic through (x̄, t̄) is x = xj (t), we have (11.18) uj (x̄, t̄) = vj (xj (0)) + ∫ t̄ 0 fj (xj (s), s) ds, and thus uj (x̄, t̄) depends on vj at only one point and on fj along the char- acteristic through (x̄, t̄), see Fig. 11.5. Consider now the case that B ̸= 0. We may then use an iterative scheme for the solution of (11.17) by setting 176 11 Hyperbolic Equations t x(x0, 0) x = xN (t) x = xN−1(t) x = x2(t) x = x1(t) Fig. 11.6. Domain of influence. u0 = 0, in R × R+, and with uk+1 definied from uk for k ≥ 0 by ∂uk+1 ∂t + A ∂uk+1 ∂x = f − Buk, in R × R+, uk+1(·, 0) = v, in R, or, in view of (11.18) (11.19) u0 = 0, uk+1j (x̄, t̄) = vj (xj (0)) + ∫ t̄ 0 (f − Buk)j (xj (s), s) ds, k ≥ 0. It is not difficult to show that the uk converge to a solution of (11.17) as k → ∞, so that the following holds (cf. Problem 7.4). Theorem 11.4. The strictly hyperbolic system (11.17) has a solution if A, B, f , and v are appropriately smooth. When A is diagonal this solution may be obtained from the iterative scheme (11.19). The uniqueness of the solution will follow from Theorem 11.5 below. We note from (11.19) and Fig. 11.5 that only the values of v in the interval (xN (0), x1(0)) enter in the successive definitions of the u k, and that f and B are only evaluated in the curvilinear triangle determined by the extreme characteristics x = xN (t) and x = x1(t). This thus determines the domain of 11.4 Symmetric Hyperbolic Systems 177 dependence of the solution at (x̄, t̄) upon the data. Similarly, the initial values at a point (x0, 0) only influence the solution for t > 0 in a wedge between
the characteristics corresponding to λ1 and λN , and originating at (x
0, 0),
see Fig. 11.6.
Example 11.9. Consider the initial value problem for the wave equation
(11.20)
∂2u
∂t2
=
∂2u
∂x2
, in R × R+,
u(·, 0) = v,
∂u
∂t
(·, 0) = w, in R.
We introduce new variables U1 = ∂u/∂t, U2 = ∂u/∂x and find for U =
(U1, U2)
T the system
∂U1
∂t
−
∂U2
∂x
= 0,
∂U2
∂t
−
∂U1
∂x
= 0,
in R × R+,
U1(·, 0) = w, U2(·, 0) = v
′, in R,
or
∂U
∂t
+ A
∂U
∂x
= 0, with U (x, 0) =
[
w(x)
v′(x)
]
,
where A =
[
0 −1
−1 0
]
. The eigenvalues of A are λ1 = −1, λ2 = 1. Setting
P =
1
√
2
[
1 1
1 −1
]
, U = P V,
we find for the new dependent variable V = (V1, V2)
T the system
∂V
∂t
+
[
−1 0
0 1
]
∂V
∂x
= 0, in R × R+,
or
∂V1
∂t
−
∂V1
∂x
= 0,
∂V2
∂t
+
∂V2
∂x
= 0.
Hence
V1(x, t) = V1(x + t, 0), V2(x, t) = V2(x − t, 0).
Going back to the original variables U , this may be used to derive d’Alembert’s
formula (11.10) for the solution of (11.20) (cf. Problem 11.5).

178 11 Hyperbolic Equations
Consider now the generalization of the system (11.17) to d space dimen-
sions,
(11.21)
∂u
∂t
+
d∑
j=1
Aj
∂u
∂xj
+ Bu = f, in Rd × R+,
u(·, 0) = v, in Rd,
where u = u(x, t) is an N -vector valued function, Aj = Aj (x, t) are symmetric
N × N matrices, B = B(x, t) an N × N matrix, and f = f (x, t) and v = v(x)
N -vectors, all of which depending smoothly and boundedly on their variables.
We also assume that solutions are small for large |x| in such a way that the
following analysis is valid.
A system such as in (11.17) is called a symmetric hyperbolic system or a
Friedrichs system. A special case is Maxwell’s equations in electro-dynamics,
see Problem 11.15. The classical wave equation utt = ∆u may be trans-
formed into a symmetric hyperbolic system by introduction of the first order
derivatives as new dependent variables. We leave the verification to Prob-
lem 11.11. More generally, many other important equations of mathematical
physics can be written as symmetric hyperbolic systems, sometimes after a
transformation of the dependent variables.
According to (11.3) the characteristic directions (ξ, τ ) = (ξ1, . . . , ξd, τ ) are
given by
det Λ(ξ, τ ) = det
(
τ I +
d∑
j=1
ξj Aj
)
= 0.
It is clear that for any given ξ this equation has N real roots τj (ξ),
j = 1, . . . , N , namely the eigenvalues of the symmetric N × N matrix
−
∑d
j=1 ξj Aj .
In general, if d > 1, it is not possible to simultaneously diagonalize the
matrices Aj , and thus the problem may not be treated as above. We shall
therefore restrict ourselves here to applying the energy method to show a
stability estimate for this problem with respect to ∥ · ∥ = ∥ · ∥L2(Rd).
Theorem 11.5. We have for the solution of (11.21), with C = C(T ),
∥u(t)∥ ≤ C
(
∥v∥ +
(∫ T
0
∥f∥2 ds
)1/2)
, for 0 ≤ t ≤ T.
Proof. We multiply the equation by u and integrate over Rd to obtain
(
∂u
∂t
, u) +
d∑
j=1
(Aj
∂u
∂xj
, u) + (Bu, u) = (f, u).
Here

11.4 Symmetric Hyperbolic Systems 179
(
∂u
∂t
, u) =
∫
Ω
⟨
∂u
∂t
, u⟩ dx = 1
2
d
dt
∥u∥2,
and
(Aj
∂u
∂xj
, u) =
∫
Rd
⟨Aj
∂u
∂xj
, u⟩ dx,
where ⟨·, ·⟩ is the standard inner product in RN . We have
∂
∂xj
⟨Aj u, u⟩ = ⟨
∂Aj
∂xj
u, u⟩ + ⟨Aj
∂u
∂xj
, u⟩ + ⟨Aj u,
∂u
∂xj
⟩,
and, since Aj is symmetric, the last two terms are equal. Further, assuming
that u is small for large |x|,
∫
Rd
∂
∂xj
⟨Aj u, u⟩ dx = 0,
and hence
(Aj
∂u
∂xj
, u) = − 1
2
(
∂Aj
∂xj
u, u).
We conclude that
1
2
d
dt
∥u∥2 + (B̃u, u) ≤ ∥f∥ ∥u∥,
where
B̃ = B − 1
2
d∑
j=1
∂Aj
∂xj
,
and hence
d
dt
∥u∥2 ≤ 2∥B̃∥C∥u∥
2 + 2∥f∥∥u∥ ≤ C0∥u∥
2 + ∥f∥2
with C0 = 2∥B̃∥C + 1. This implies
∥u(t)∥2 ≤ ∥v∥2 +
∫ T
0
∥f∥2 ds + C0
∫ t
0
∥u∥2 ds, for 0 ≤ t ≤ T,
so that by Gronwall’s lemma (cf. Problem 7.6),
∥u(t)∥2 ≤ eC0T
(
∥v∥2 +
∫ T
0
∥f∥2 ds
)
, for 0 ≤ t ≤ T.
⊓.
In the usual way, this inequality implies uniqueness and stability for the
problem (11.21). Existence of a solution may be shown, for instance, by con-
structing a finite difference approximation on a mesh with mesh-width h and
then showing convergence as h → 0.

180 11 Hyperbolic Equations
It is also possible to show here that, as in the case of one space dimension
treated above, the value of the solution of (11.21) at a point (x̄, t̄) with t̄ > 0
only depends on data in a finite domain. To do so we consider for simplicity
the case of a homogeneous equation with constant coefficients and with no
lower order term, so that the problem is
∂u
∂t
+
d∑
j=1
Aj
∂u
∂xj
= 0, in R × R+,
u(·, 0) = v, in Rd.
Then the characteristic polynomial is the symmetric matrix
Λ(ξ, τ ) = τ I +
d∑
j=1
ξj Aj .
Consider now a circular cone K with vertex (x̄, t̄), restricted to t ≤ t̄,
cf. (11.9), and with such an opening angle that the exterior unit normal
(nx, nt) on the mantle M of the cone makes Λ(nx, nt) positive definite. That
it is possible to find such a cone follows from the fact that for the direction
(0,1) we have Λ(0, 1) = I which is positive definite, and hence Λ(ξ, 1) is also
positive definite for small |ξ|.
Let B0 be the domain in the plane t = 0 cut out by the cone. We claim
that if v = 0 in B0, then u(x̄, t̄) = 0.
To prove this we use again the energy method. We multiply the equation
by u and integrate over K, using the assumption that the Aj are symmetric
and constant, to obtain
0 =
∫
K
(
⟨
∂u
∂t
, u⟩ +
d∑
j=1
⟨Aj
∂u
∂xj
, u⟩
)
dx dt
= 1
2
∫
K
( ∂
∂t
⟨u, u⟩ +
d∑
j=1
∂
∂xj
⟨Aj u, u⟩
)
dx dt.
By the divergence theorem we have
∫
M
(
⟨u, u⟩ nt +
d∑
j=1
⟨Aj u, u⟩ nxj
)
ds =
∫
B0
⟨u, u⟩ dx,
or, since u = 0 in B0,
∫
M
⟨Λ(nx, nt)u, u⟩ ds = 0,
which implies u = 0 on M , since Λ(nx, nt) is positive definite. In particular,
u(x̄, t̄) = 0, which is our claim.

11.5 Problems 181
11.5 Problems
Problem 11.1. Determine the characteristics for the Tricomi equation
∂2u
∂x21
+ x1
∂2u
∂x22
= f, for x = (x1, x2) ∈ R
2.
Problem 11.2. Find the characteristic directions of the Cauchy-Riemann
equations
∂u
∂x
−
∂v
∂y
= 0,
∂u
∂y
+
∂v
∂x
= 0.
Problem 11.3. Show (11.7) directly from (11.6).
Problem 11.4. Let u be as in (11.6) and assume that v ∈ H2 ∩H10 , w ∈ H
1
0 .
Show that
∥u(t)∥ ≤ C
(
∥v∥ + ∥w∥
)
,
∥∇u(t)∥ ≤ C
(
∥∇v∥ + ∥w∥
)
,
∥utt(t)∥ = ∥∆u(t)∥ ≤ C
(
∥∆v∥ + ∥∇w∥
)
,
∥u(t) − v∥ ≤ Ct
(
∥∇v∥ + ∥w∥
)
,
∥ut(t) − w∥ ≤ Ct
(
∥∆v∥ + ∥∇w∥
)
.
Hence u is a solution of (11.5) at least in the L2 sense. Hint: Recall Theorem
6.4 and Problem 6.3. Show that
∥u(t) − v∥2 = t2
∞∑
j=1
(√
λj v̂j
cos(
√
λj t) − 1√
λj t
+ ŵj
sin(
√
λj t)√
λj t
)2
.
Problem 11.5. Prove d’Alembert’s solution formula (11.10) for the Cauchy
problem for the one-dimensional wave equation, i.e.,
utt − uxx = 0 in R × R+,
u(·, 0) = v, ut(·, 0) = w, in R.
Problem 11.6. (a) Solve the initial-value problem
∂u
∂t
+
[
0 x
x 0
]
∂u
∂x
= 0, x ∈ R, t > 0,
u(x, 0) = v(x), x ∈ R,
by the method of characteristics.
(b) Prove a stability estimate by the energy method.
Problem 11.7. (a) Solve the initial value problem
ut + (x + t)ux = 0 for (x, t) ∈ R × R+, with u(x, 0) = v(x) for x ∈ R,

182 11 Hyperbolic Equations
by means of the method of characteristics.
(b) Show that
∥u(·, t)∥ = et/2∥v∥ and ∥ux(·, t)∥ = e
−t/2
∥vx∥, for t ≥ 0,
by the energy method. Check these results by a direct calculation using the
solution formula from (a).
Problem 11.8. Solve the problem
x1
∂u
∂x1
− x2
∂u
∂x2
= 0 for x ∈ R2, with u(x) = ϕ(x) for x ∈ S,
where S is a non-characteristic curve.
Problem 11.9. Solve the problem
x1
∂u
∂x1
+ 2×2
∂u
∂x2
+
∂u
∂x3
= 3u for x ∈ R3,
u(x1, x2, 0) = ϕ(x1, x2) for (x1, x2) ∈ R
2.
Problem 11.10. Prove the following stability estimate for the problem
(11.11), (11.13) under a suitable condition on the coefficients aj :
∫
Ω
u2 dx +
∫
Γ+
u2 n · a ds ≤ C
(∫
Ω
f 2 dx +
∫
Γ−
v2|n · a| ds
)
.
Problem 11.11. Show that the wave equation utt−∆u = 0 can be written as
a symmetric hyperbolic system by introduction of the first order derivatives
ut, ux1 , . . . , uxd as new dependent variables.
Problem 11.12. Modify the proof of Theorem 11.5 to show the slightly
stronger result
∥u(t)∥ ≤ C(T )
(
∥v∥ +
∫ T
0
∥f (s)∥ ds
)
, for 0 ≤ t ≤ T.
Problem 11.13. In addition to the assumptions of Theorem 11.5 assume
that Aj are constant and B symmetric positive semidefinite. Prove
∥u(t)∥ ≤ ∥v∥ +
∫ t
0
∥f∥ ds, for t ≥ 0.
Problem 11.14. Generalize Theorem 11.5 to symmetric hyperbolic systems
of the form
M
∂u
∂t
+
d∑
j=1
Aj
∂u
∂xj
+ Bu = f, in Rd × R+,
where Aj and B are as before and M = M (x, t) is symmetric positive definite
uniformly with respect to x, t, so that ⟨M (x, t)ξ, ξ⟩ ≥ α |ξ|2 for all ξ ∈ RN ,
(x, t) ∈ Rd × R+, with α > 0.

11.5 Problems 183
Problem 11.15. The evolution of the electric field E(x, t) ∈ R3 and mag-
netic field H(x, t) ∈ R3 in a homogeneous and isotropic space can be de-
scribed by the following two of Maxwell’s equations (Ampère’s law and Fara-
day’s law)
(11.22)
1
c
∂E
∂t
− ∇ × H +
4π
c
J = 0, in R3 × R+,
1
c
∂H
∂t
+ ∇ × E = 0, in R3 × R+,
where c is a positive constant and
∇ × H = curl H =
(∂H3
∂x2
−
∂H2
∂x3
,
∂H1
∂x3
−
∂H3
∂x1
,
∂H2
∂x1
−
∂H1
∂x2
)
.
Let us also assume that the density of current J satisfies Ohm’s law J = σE
with σ a nonnegative constant. Show that (11.22) with E and H given at
t = 0 constitute a well posed problem by showing that (11.22) is a Friedrichs
system. What can be said about the stability of the energy density e =
1
2
(E · E + H · H)? Hint: Problem 11.13.
Problem 11.16. Recall the equation
ρ
∂2u
∂t2
=
∂
∂x
(
E
∂u
∂x
)
for the longitudinal motion of an elastic bar from Problem 1.2.
(a) Assume for simplicity that ρ and E are constant and show that it can be
written as a symmetric hyperbolic system
[
ρ 0
0 E
]
Ut −
[
0 E
E 0
]
Ux = 0
in the variables U1 = ut, U2 = ux, cf. Problem 11.14.
(b) Assume, e.g., the boundary conditions u(0) = 0, ux(L) = 0. Show that
the mechanical energy is conserved, i.e., with e = 1
2
(ρu2t + Eu
2
x),
∫ L
0
e(x, t) dx =
∫ L
0
e(x, 0) dx.
Problem 11.17. Compute the eigenvalues and normalized eigenvectors of
the symmetric matrix A(x, t) =
[
x t
t −x
]
. Show that the eigenvector matrix
P (x, t) is discontinuous at x = 0, t = 0, where the eigenvalues are multiple.

12 Finite Difference Methods for Hyperbolic
Equations
Solution of hyperbolic equations is perhaps the area in which finite difference
methods have most successfully continued to play an important role. This is
particularly true for nonlinear conservation laws, which, however, are beyond
the scope of this elementary presentation. Here we begin in Sect. 12.1 with
the pure initial-value problem for a first order scalar equation in one space
variable and study stability and error estimates for the basic upwind scheme,
the Friedrichs scheme, and the Lax-Wendroff scheme. In Sect. 12.2 we extend
these considerations to symmetric hyperbolic systems and also to higher space
dimension, and in Sect. 12.3 we treat the Wendroff box scheme for a mixed
initial-boundary value problem in one space dimension.
12.1 First Order Scalar Equations
In this first section we consider the simple model initial value problem
(12.1)
∂u
∂t
= a
∂u
∂x
, in R × R+,
u(·, 0) = v, in R,
where a is a constant. We recall that for v ∈ C1 this problem admits the
unique classical solution
(12.2) u(x, t) = (E(t)v)(x) = v(x + at),
which may thus be found by following the characteristic x + at = constant
through (x, t) backwards to t = 0 and taking the value of v at that point. A
similar statement holds for variable coefficient a = a(x), in which case the
characteristic is curved. Since the solution operator just affects a shift of the
argument both the maximum-norm and the L2-norm are constant in time,
(12.3) ∥E(t)v∥C = ∥v∥C and ∥E(t)v∥ = ∥v∥, for t ≥ 0,
and thus, in particular, E(t) is stable in both norms.
For the purpose of solving the model problem approximately by the finite
difference method we introduce, as earlier for parabolic equations in Sect. 9.1,

186 12 Finite Difference Methods for Hyperbolic Equations
a mesh size h in space and a time step k and denote the approximation
of u(x, t) at (xj , tn) = (jh, nk) by U
n
j , for j, n ∈ Z, n ≥ 0. Here Z =
{. . . , −1, −2, 0, 1, 2, . . . } is the set of all integers. Assuming that a > 0 we
replace (12.1) by
(12.4)
∂tU
n
j = a∂xU
n
j , for j, n ∈ Z, n ≥ 0,
U 0j = Vj = v(xj ), for j ∈ Z,
where as earlier ∂t and ∂x denote forward difference quotients, so that the
difference equation reads
U n+1j − U
n
j
k
= a
U nj+1 − U
n
j
h
.
Introducing this time the mesh ratio λ = k/h, which we assume is kept
constant as h and k tend to zero, we see that (12.4) is an explicit scheme,
which defines the approximation at t = tn+1 by
(12.5) U n+1j = (EkU
n)j = aλU
n
j+1 + (1 − aλ)U
n
j , for j, n ∈ Z, n ≥ 0.
If we think of U n as being defined for all x in R and not only at the mesh
points x = xj , we may write
(12.6) U n+1(x) = (EkU
n)(x) = aλU n(x + h) + (1 − aλ)U n(x), x ∈ R.
By iteration we find for the approximate solution at t = tn
U n(x) = (Enk v)(x), for x ∈ R.
Similarly to the situation for the heat equation we find that Ek is stable
in maximum-norm if aλ ≤ 1, since the coefficients of Ek are then positive
and add up to 1, so that
∥Ekv∥C ≤ ∥v∥C,
and hence also
∥U n∥C = ∥E
n
k v∥C ≤ ∥v∥C.
It is also easy to see that the condition aλ ≤ 1 is necessary for stability. As
earlier stability implies convergence:
Theorem 12.1. Let U n and u be defined by (12.6) and (12.1), and assume
that 0 < aλ ≤ 1. Then ∥U n − un∥C ≤ Ctnh|v|C2 , for tn ≥ 0. Proof. We introduce the truncation error (12.7) τ n(x) := ∂tu n(x) − a∂xu n(x), 12.1 First Order Scalar Equations 187 and find by Taylor expansion for an exact solution u of the differential equa- tion, with In = (tn, tn+1), |τ n(x)| ≤ |∂tu n(x) − ut(x, tn)| + a|∂xu n(x) − aux(x, tn)| ≤ C(h + k) max t∈In ( |utt(·, t)| + |uxx(·, t)| ) ≤ Ch|v|C2 , (12.8) where we have used that k ≤ λh, that utt = auxx, and that |uxx(·, t)|C ≤ |v|C2 . We may also write (12.7) in the form un+1(x) = Eku n(x) + kτ n(x), for x ∈ R. Setting zn = U n − un we therefore have zn+1 = Ekz n − kτ n, or, by repeated application, zn = Enk z 0 − k n−1∑ j=0 E n−1−j k τ j . Since z0 = U 0 − u0 = v − v = 0, we conclude by stability and (12.8), ∥zn∥C ≤ k n−1∑ j=0 ∥τ j∥C ≤ C nk h|v|C2 , for tn ≥ 0, which completes the proof of the theorem. ⊓. Note that if a < 0, the natural choice of finite difference approximation is, instead of (12.4), (12.9) ∂tU n j = a∂̄xU n j , for n ≥ 0, or, cf. (12.5), U n+1j = (EkU n)j = −aλU n j−1 + (1 + aλ)U n j , for j ∈ Z, n ≥ 0. The stability condition is now 0 < −aλ ≤ 1. Since both (12.4) and (12.9) use points in the direction of the flow, these difference schemes are referred to as upwind schemes. Let us consider more generally an explicit finite difference scheme (12.10) U n+1j = (EkU n)j = ∑ p apU n j−p, for j, n ∈ Z, n ≥ 0, where ap = ap(λ) with λ = k/h = const., or, with x allowed to vary over R, 188 12 Finite Difference Methods for Hyperbolic Equations (12.11) U n+1(x) = (EkU n)(x) = ∑ p apU n(x − ph), for x ∈ R, n ≥ 0, U 0(x) = v(x), for x ∈ R. We say that such a method is accurate of order r if τ n = k−1(un+1 − Eku n) = O(hr), as h → 0, where u is the exact solution of (12.1) and k/h = λ = constant. We note that as in Sect. 9.1 we have for the Fourier transform of Ekv (Ekv)̂ (ξ) = Ẽ(hξ)v̂(ξ), where Ẽ(ξ) = ∑ p ape −ipξ, and, in exactly the same way as in the parabolic case, a necessary and suffi- cient condition for stability in L2 is the von Neumann condition (12.12) |Ẽ(ξ)| ≤ 1, for ξ ∈ R. For our above scheme (12.5) we have Ẽ(ξ) = aλeiξ + 1 − aλ, and as ξ varies, Ẽ(ξ) belongs to a circle in the complex plane with center at 1 − aλ and radius aλ. In order for (12.12) to hold it is therefore necessary and sufficient that aλ ≤ 1, which is our old stability condition. In the same way as for the parabolic problem in Sect. 9.1, the definition of the accuracy of the method may also be expressed in terms of the trigono- metric polynomial Ẽ(ξ): The method is accurate of order r if and only if (12.13) Ẽ(ξ) = eiaλξ + O(ξr+1), as ξ → 0, where in the proof we use that for the exact solution we have (E(t)v)̂ (ξ) = ∫ R v(x + at)e−ixξ dx = eiatξv̂(ξ). As in Sect. 9.1 one may then prove the following error estimate. Theorem 12.2. Let U n and u be defined by (12.11) and (12.1), and assume Ek is accurate of order r and stable in L2. Then ∥U n − un∥ ≤ Ctnh r |v|r+1, for tn ≥ 0. Another natural choice for a difference approximation to (12.1) is obtained by replacing the derivative with respect to x by the symmetric difference quotient ∂̂xU n(x) = U n(x + h) − U n(x − h) 2h , 12.1 First Order Scalar Equations 189 which results in the finite difference equation (12.14) U n+1(x) − U n(x) k = a U n(x + h) − U n(x − h) 2h , and thus in the difference scheme U n+1(x) = (EkU n)(x) = U n(x) + 1 2 aλ(U n(x + h) − U n(x − h)), n ≥ 0, U 0(x) = v(x), x ∈ R. In this case the symbol of Ek is Ẽ(ξ) = 1 + 1 2 aλ(eiξ − e−iξ) = 1 + aλi sin ξ. Since |Ẽ(ξ)|2 = 1 + a2λ2 sin2 ξ > 1, except at ξ = mπ,
we conclude that this method is unstable for any choice of λ.
The latter scheme may be stabilized by replacing U n(x) on the left in
(12.14) by the average 1
2
(U n(x + h) + U n(x − h)), which results in
U n+1(x) − 1
2
(U n(x + h) + U n(x − h))
k
= a
U n(x + h) − U n(x − h)
2h
,
or
U n+1(x) = (EkU
n)(x) = 1
2
(1 + aλ)U n(x + h) + 1
2
(1 − aλ)U n(x − h).
This is a special case of the Friedrichs scheme which we shall study in more
generality below. Here
Ẽ(ξ) = cos ξ + iaλ sin ξ,
and we find
|Ẽ(ξ)|2 = cos2 ξ + a2λ2 sin2 ξ ≤ 1, for ξ ∈ R,
if and only if |aλ| ≤ 1.
This case of the Friedrichs scheme may also be written in the form
U n+1(x) − U n(x)
k
= a
U n(x + h) − U n(x − h)
2h
+
1
2k
(
U n(x + h) − 2U n(x) + U n(x − h)
)
,
or
(12.15) ∂tU
n = a∂̂xU
n +
1
2
h
λ
∂x∂̄xU
n.
This equation may be thought of as an approximation to a parabolic equation
with the small diffusion coefficient 1
2
h/λ. The stability of this scheme may

190 12 Finite Difference Methods for Hyperbolic Equations
be interpreted as the result of introducing artificial diffusion in the original
scheme (12.14). (This is also referred to as artificial viscosity in computational
fluid dynamics.)
Let us note that for the Friedrichs scheme U n(x) may be expressed in
terms of the initial data in the form
U n(x) =
n∑
j=−n
anj v(x − jh),
and thus uses the values of v(x) in the interval [x − nh, x + nh] = [x −
tn/λ, x + tn/λ]. The exact solution at t = tn is given by (12.2) as the value
of v at x + tna. It is clear that if the domain of dependence of the difference
scheme, i.e., the interval [x − tn/λ, x + tn/λ], does not contain the domain
of dependence of the exact solution, namely the point x + tna, then the
difference method could not possibly be successful. This condition reduces to
−1 ≤ aλ ≤ 1, which is our old stability criterion.
For a general scheme of the form (12.10) we may thus formulate the
Courant-Friedrichs-Lewy condition (or the CFL condition) for stability: In
order for the scheme to be stable it is necessary that the domain of dependence
of the finite difference scheme at (x, t) contains the domain of dependence of
the continuous problem.
In our first example (12.4) we find that the finite difference scheme has the
interval of dependence [x, x + tn/λ] and thus that the CFL condition requires
0 ≤ aλ ≤ 1. In particular, we recover our old stability condition aλ ≤ 1,
and also note that the scheme (12.4) cannot be used for a < 0. In the same way we find that for a > 0 the forward difference quotient in (12.4) could
not successfully be replaced by a backward difference quotient. For a < 0, however, as we have learned, the scheme (12.4) with ∂x replaced by ∂̄x is stable if aλ ≥ −1. That the CFL condition is not sufficient for stability is shown by the scheme (12.14), which has the same domain of dependence as the Friedrichs scheme but which is unstable for all λ. Like our first example (12.4) the Friedrichs scheme is also first order ac- curate: If the exact solution u of (12.1) is sufficiently regular, then we have by the representation (12.15) that ∂tu n − a∂̂xu n − 1 2 h λ ∂x∂̄xu n = unt + 1 2 kuntt − au n x − 1 2 h λ unxx + O(h 2) = 1 2 h λ (λ2untt − u n xx) + O(h 2) = 1 2 h λ (a2λ2 − 1)unxx + O(h 2), as h → 0. Thus the error is first order except for the special choice λ = 1/|a|, in which case the approximate solution is equal to the exact solution (cf. (12.13)). Next we propose to determine a second order accurate scheme of the form 12.1 First Order Scalar Equations 191 U n+1(x) = (EkU n)(x) = a1U n(x − h) + a0U n(x) + a−1U n(x + h). The formula (12.13) shows that the condition for this is a1e −iξ + a0 + a−1e iξ = eiaλξ + O(ξ3), as ξ → 0, or, by Taylor expansion, (a1 + a0 + a−1) − i(a1 − a−1)ξ − 1 2 (a1 + a−1)ξ 2 = 1 + iaλξ − 1 2 a2λ2ξ2 + O(ξ3), as ξ → 0, that is, a1 + a0 + a−1 = 1, a1 − a−1 = −aλ, a1 + a−1 = a 2λ2. This results in a−1 = 1 2 (a2λ2 + aλ), a0 = 1 − a 2λ2, a1 = 1 2 (a2λ2 − aλ), and thus (EkU n)(x) = 1 2 (a2λ2 + aλ)U n(x + h) + (1 − a2λ2)U n(x) + 1 2 (a2λ2 − aλ)U n(x − h), which gives Ẽ(ξ) = 1 − a2λ2 + a2λ2 cos ξ + iaλ sin ξ. We find by a simple calculation (12.16) |Ẽ(ξ)|2 = 1 − a2λ2(1 − a2λ2)(1 − cos ξ)2, and hence the method is stable in L2 exactly if a 2λ2 ≤ 1, see Problem 12.2. Again this agrees with the CFL necessary condition for stability. This latter method is referred to as the Lax-Wendroff method. We remark that this is an example of a method which is not stable in maximum-norm even though it is L2-stable. In fact, it can be shown that, if 0 < a 2λ2 < 1, then ∥Enk v∥C ≤ Cn 1/12 ∥v∥C, and that this estimate is sharp in terms of the power of n. However, this power is small and the effect of the instability is in general not noticeable. 192 12 Finite Difference Methods for Hyperbolic Equations 12.2 Symmetric Hyperbolic Systems Much of what has been said in Sect. 12.1 generalizes to systems in one space dimension, ∂u ∂t = A ∂u ∂x , in R × R+, where u = (u1, . . . , uN ) T is a vector with N components and A is a symmetric N × N matrix. For instance, the Friedrichs scheme now takes the form (12.17) U n+1(x) = (EkU n)(x) = 1 2 (I + λA)U n(x + h) + 1 2 (I −λA)U n(x−h) and the Lax-Wendroff scheme is U n+1(x) = (EkU n)(x) = 1 2 (λ2A2 + λA)U n(x + h) + (I − λ2A2)U n(x) + 1 2 (λ2A2 − λA)U n(x − h). (12.18) The symbols of these operators now become the matrix-valued periodic func- tions Ẽ(ξ) = I cos ξ + iλA sin ξ, and Ẽ(ξ) = I − λ2A2 + λ2A2 cos ξ + iλA sin ξ, respectively. These may be diagonalized by the same orthogonal transforma- tion as A and one finds easily that the stability requirement in both cases is that |A|λ ≤ 1, where |A| is the matrix norm subordinate to the Euclidean norm on RN , i.e., |A| = sup v ̸=0 |Av| |v| = max j=1,...,N |µj|, where µj are the eigenvalues of A. For an example of such a system we consider the initial value problem (12.19) ∂2w ∂t2 = a2 ∂2w ∂x2 , in R × R+, w(·, 0) = w0, ∂w ∂t (·, 0) = w1, in R. The second order hyperbolic equation may be reduced to a system by setting u1 = a ∂w ∂x , u2 = ∂w ∂t . These functions then satisfy 12.2 Symmetric Hyperbolic Systems 193 ∂u1 ∂t = a ∂u2 ∂x , ∂u2 ∂t = a ∂u1 ∂x , i.e., we have for u = (u1, u2) T, (12.20) ∂u ∂t = [ 0 a a 0 ] ∂u ∂x , in R × R+, u(·, 0) = [ aw′0 w1 ] , in R. Conversely, the solution of (12.20) determines the solution of (12.19). Either of the two schemes (12.17) and (12.18) may now be applied to the present system. Since the matrix in (12.20) has eigenvalues ±a we rediscover our standard stability criterion |a|λ ≤ 1. We briefly turn to the case of more than one space dimension and consider a symmetric hyperbolic system (or Friedrichs system), (12.21) ∂u ∂t = d∑ j=1 Aj ∂u ∂xj , in R × R+, u(·, 0) = v, in Rd, where u is an N -vector valued function and the Aj are symmetric N ×N ma- trices. We recall from Sect. 11.4 that the corresponding initial value problem is correctly posed in L2. We consider now an associated finite difference operator (12.22) U n+1(x) = (EkU n)(x) = ∑ β aβ U n(x − βh), where β = (β1, . . . , βd) has integer components and the aβ are finitely many constant N × N matrices. With ξ = (ξ1, . . . , ξd) ∈ R d the symbol is now the matrix Ẽ(ξ) = ∑ β aβ e −iβ·ξ, where β · ξ = β1ξ1 + · · · + βdξd. After Fourier transformation we have now (U n+1)̂ (ξ) = Ẽ(hξ)(U n)̂ (ξ), for n ≥ 0, and hence (U n)̂ (ξ) = Ẽ(hξ)nv̂(ξ). It follows easily that a necessary and sufficient condition for stability in L2 is that the matrix norm of the symbol satisfies (12.23) |Ẽ(ξ)n| ≤ C, for n ≥ 0, ξ ∈ Rd. 194 12 Finite Difference Methods for Hyperbolic Equations In contrast to the scalar case it is not true for matrices that |An| ≤ C for n ≥ 0 implies |A| ≤ 1, as the example (12.24) [ a 1 0 a ]n = [ an nan−1 0 an ] , with |a| < 1 shows. However, if (12.23) holds, it follows that for each eigen- value λj (ξ) of Ẽ(ξ) we have |λj (ξ) n| ≤ C, for ξ ∈ Rd, n ≥ 0, and hence |λj (ξ)| ≤ 1, for ξ ∈ R d. This is referred to as von Neumann’s stability condition, and is thus a neces- sary condition for stability in L2. It is not a sufficient condition as illustrated by the example (12.24) with a = 1. A sufficient condition for stability in L2 is obviously |Ẽ(ξ)| ≤ 1, for ξ ∈ Rd. In order to be able to construct a stable difference scheme of the above form we shall need the following result. Lemma 12.1. Assume that aβ are symmetric, positive semidefinite matrices with ∑ β aβ = I. Then |Ẽ(ξ)| = ∣ ∣ ∣ ∑ β aβ e −iβξ ∣ ∣ ∣ ≤ 1, for ξ ∈ Rd. Proof. Let ⟨u, v⟩ = N∑ j=1 uj vj , for u, v ∈ C N . Since aβ is real, symmetric and positive semidefinite we see that the corre- sponding bilinear form ⟨aβ u, v⟩ satisfies ⟨aβ u, v⟩ = ⟨aβ v, u⟩, ⟨aβ u, u⟩ ≥ 0, for u, v ∈ C N . Using these properties it is easy to prove the generalized Cauchy-Schwarz inequality |⟨aβ u, v⟩| ≤ ⟨aβ u, u⟩ 1/2 ⟨aβ v, v⟩ 1/2. (The proof of the standard Cauchy-Schwarz inequality works; this inequality is a generalization because ⟨aβ u, u⟩ 1/2 is only a seminorm.) Hence, using also the inequality 2ab ≤ a2 + b2, we have |⟨aβ u, v⟩| ≤ 1 2 ⟨aβ u, u⟩ + 1 2 ⟨aβ v, v⟩. Therefore, 12.2 Symmetric Hyperbolic Systems 195 |⟨Ẽ(ξ)v, w⟩| ≤ ∑ β |⟨aβ e −iβ·ξv, w⟩| ≤ 1 2 ∑ β ⟨aβ v, v⟩ + 1 2 ∑ β ⟨aβ w, w⟩ = 1 2 |v|2 + 1 2 |w|2. Taking w = Ẽ(ξ)v, we conclude |w|2 ≤ 1 2 |v|2 + 1 2 |w|2, which completes the proof. ⊓. As an application, we take the Friedrichs scheme (of which we have seen particular cases above) U n+1(x) = (EkU n)(x) = 1 2 d∑ j=1 {(1 d I + λAj ) U n(x + hej ) + (1 d I − λAj ) U n(x − hej ) } , (12.25) where ej is the unit vector in the direction of xj . This may also be written, similarly to (12.15), ∂tU n = d∑ j=1 Aj ∂̂xj U n + 1 2 h λd d∑ j=1 ∂xj ∂̄xj U n, and is thus, in particular, consistent with (12.21). Now, if λ is such that (12.26) 0 < λ ≤ min 1≤j≤d (d|Aj|) −1, then the coefficients of (12.25) are positive semidefinite and hence the lemma shows stability in L2. For instance, for the system ∂u ∂t = [ 1 0 0 −1 ] ∂u ∂x1 + [ 0 1 1 0 ] ∂u ∂x2 , the condition (12.26) reduces to 0 < λ ≤ 1/2. In this particular case Ẽ(ξ) = 1 2 I(cos ξ1 + cos ξ2) + λi [ sin ξ1 sin ξ2 sin ξ2 − sin ξ1 ] , which is a normal matrix (one which commutes with its adjoint). For such a matrix its norm equals the maximum modulus of its eigenvalues, and using this one may prove that |Ẽ(ξ)| ≤ 1 for λ ≤ 1/ √ 2, which is less restrictive a condition in this case than the above condition based on the lemma. The Friedrichs scheme is again only first order accurate. In fact, it may be shown that schemes of the form (12.22) with aβ positive semidefinite may in general only be accurate of first order. 196 12 Finite Difference Methods for Hyperbolic Equations 12.3 The Wendroff Box Scheme In this final section we shall describe the second order Wendroff box scheme, which is suitable for mixed initial-boundary value problems and also for sys- tems in the case of one space dimension. We consider thus the initial boundary value problem ∂u ∂t + a ∂u ∂x + bu = f, in Ω × J, where Ω = (0, 1), J = (0, T ), u(0, ·) = g, on J, u(·, 0) = v, in Ω, where a, b, f, g, and v are smooth functions with a positive and where g(0) = v(0) for compatibility at (x, t) = (0, 0). Note that since a is positive the boundary values have been prescribed on the left boundary. With U nj the value of the mesh function at (xj , tn) = (jh, nk), 0 ≤ j ≤ M, 0 ≤ n ≤ N , where M h = 1, N k = T , we define also Uj+1/2 = 1 2 (Uj + Uj+1) and U n+1/2 = 1 2 (U n + U n+1), and U n+1/2 j+1/2 = 1 4 (U nj + U n+1 j + U n j+1 + U n+1 j+1 ). The Wendroff box scheme is then (12.27) ∂tU n j+1/2 + a ∂xU n+1/2 j + bU n+1/2 j+1/2 = f, 0 ≤ j < M, 0 ≤ n < N, U n0 = G n = g(tn), 0 ≤ n ≤ N, U 0j = Vj = v(xj ), 0 ≤ j ≤ M, where a, b, and f are evaluated at (xj+1/2, tn+1/2). The difference equation may also be written U n+1j + U n+1 j+1 − U n j − U n j+1 2k + a U n+1j+1 + U n j+1 − U n+1 j − U n j 2h + b U n+1j + U n+1 j+1 + U n j + U n j+1 4 = f ; (12.28) and we see by symmetry that it is second order accurate. This equation may also be expressed as (1 + aλ+ 1 2 bk)U n+1j+1 = (1 + aλ − 1 2 bk)U nj + (1 − aλ − 1 2 bk)U nj+1 − (1 − aλ + 1 2 bk)U n+1j + 2kf, where λ = k/h. (12.29) This defines the solution at (xj+1, tn+1) in terms of the values at (xj , tn), (xj+1, tn) and (xj , tn+1). Given U n one may therefore find U n+1 explicitly in the order U n+10 = G n+1, U n+11 , U n+1 2 , . . . , U n+1 M . 12.3 The Wendroff Box Scheme 197 We shall show the stability of this method in the discrete L2-norm ∥V ∥h = ( h M∑ j=1 V 2j )1/2 , and restrict ourselves, for simplicity only, to the case that a is constant and b = f = g = 0. In this case (12.28) reduces to U n+1j+1 = U n j + 1 − aλ 1 + aλ (U nj+1 − U n+1 j ), and we shall show (12.30) ∥U n∥h ≤ C∥V ∥h, for 0 ≤ tn ≤ T. For this purpose we multiply (12.27) by U n+1/2 j+1/2 and note that ∂tU n j+1/2 U n+1/2 j+1/2 = 1 2 ∂t(U n j+1/2) 2, and ∂xU n+1/2 j U n+1/2 j+1/2 = 1 2 ∂x(U n+1/2 j ) 2, so that we obtain ∂t(U m j+1/2) 2 + a∂x(U m+1/2 j ) 2 = 0. After summation over j = 0, . . . , M − 1, m = 0, . . . , n − 1 and multiplication by hk this yields, for n ≤ N, h M−1∑ j=0 (U nj+1/2) 2 + ak n−1∑ m=0 (U m+1/2 M ) 2 = h M−1∑ j=0 (U 0j+1/2) 2 + ak n−1∑ m=0 (U m+1/2 0 ) 2, which implies, since we have assumed U m+1/2 0 = 0, (12.31) h M−1∑ j=0 (U nj + U n j+1) 2 ≤ C∥V ∥2h. Similarly, multiplying (12.27) instead by hk∂x∂tU n j = U n+1 j+1 − U n+1 j − U n j+1 + U n j , we obtain after a simple calculation (12.32) h M−1∑ j=0 (U nj+1 − U n j ) 2 ≤ C∥V ∥2h. Together, (12.31) and (12.32) complete the proof of (12.30). The stability and the second order of accuracy imply second order con- vergence, provided u is smooth enough, i.e., ∥U n − un∥h ≤ C(u)h 2, for k/h = λ = constant. 198 12 Finite Difference Methods for Hyperbolic Equations 12.4 Problems Problem 12.1. Prove that the Friedrichs scheme for (12.1) is stable in the maximum norm if and only if |a|λ ≤ 1. Problem 12.2. Show (12.16) and hence that the Lax-Wendroff scheme is stable in L2 if and only if |a|λ ≤ 1. Problem 12.3. Let Ek be an explicit finite difference operator defined by (EkV )j = ∑ p apVj−p. (a) Show that ∥EkV ∥∞,h ≤ C∥v∥∞,h, with C = ∑ p |ap|, and that this inequality does not hold with any smaller constant. (b) Show that (Enk V )j = ∑ p anpVj−p, where anp = 1 2π ∫ π −π Ẽ(ξ)neipξ dξ, with Ẽ(ξ) = ∑ j aj e −ijξ the symbol of Ek. (c) Show that Ek is maximum-norm stable if and only if ∑ p |anp| ≤ C, ∀n ≥ 0. Problem 12.4. Prove that the Lax-Wendroff scheme is unstable in the max- imum norm if |a|λ > 1. Hint: Problem 12.3.
Problem 12.5. Recall from Sect. 7.1 that for an m × m matrix M one can
define exp(M ) =
∑∞
j=0
1
j!
M j . Consider the symmetric hyperbolic system
∂u/∂t = A∂u/∂x.
(a) Show that a finite difference scheme for this system of the form (cf. (12.17)
and (12.18))
(EkV )j =
∑
p
ap(λA)Vj−p
is accurate of order r, r = 1, 2, if
Ẽ(ξ) = exp(iλAξ) + O(ξr+1), as ξ → 0.
(b) Check this condition for the Friedrichs and Lax-Wendroff schemes (12.17)
and (12.18).

12.4 Problems 199
Problem 12.6. Discuss the meaning of the CFL condition for the initial-
boundary value problem in Sect. 12.3 and show that it is satisfied for the
Wendroff box scheme defined in (12.27).
Problem 12.7. (Computer exercise.) Apply the Wendroff box scheme to the
problem in Example 11.8 with h = k = 1/10 and h = k = 1/20. Calculate
the errors at (1, 1/2).

13 The Finite Element Method for Hyperbolic
Equations
In this chapter we apply the finite element method to hyperbolic equations.
In Sect. 13.1 we study an initial-boundary value problem for the wave equa-
tion, and discuss semidiscrete and completely discrete schemes based on the
standard finite element discretization in the spatial variables. In Sect. 13.2
we consider a scalar partial differential equation of first order in two indepen-
dent variables. We begin by treating the equation as an evolution equation
and show a nonoptimal order O(h) error estimate for the standard Galerkin
method. Looking instead of the associated boundary value problem as a
two-dimensional problem of the type treated in Sect. 11.3, we introduce the
streamline diffusion modification and demonstrate a O(h3/2) convergence es-
timate. We finally return to the evolution aspect and combine streamline
diffusion with the so-called discontinuous Galerkin method to design a time
stepping scheme by using two-dimensional approximating functions which
may be discontinuous at the time levels.
13.1 The Wave Equation
In this section we briefly discuss some results concerning semidiscrete and
completely discrete schemes for the following initial-boundary value problem
for the wave equation,
(13.1)
utt − ∆u = f, in Ω × R+,
u = 0, on Γ × R+,
u(·, 0) = v, ut(·, 0) = w, in Ω.
As often earlier we assume that Ω ⊂ R2 is a bounded convex domain whose
boundary Γ is a polygon, and denote by Sh ⊂ H
1
0 a family of spaces of
piecewise linear finite element functions in the spatial variables.
The semidiscrete analogue of (13.1) is then to find uh(t) ∈ Sh such that,
with our previous notation, in particular with a(v, w) = (∇v, ∇w),
(13.2)
(uh,tt, χ) + a(uh, χ) = (f, χ), ∀χ ∈ Sh, for t > 0,
uh(0) = vh, uh,t(0) = wh.

202 13 The Finite Element Method for Hyperbolic Equations
This is an initial value problem for a system of ordinary differential equations
of second order for the coefficients of uh with respect to the standard basis
{Φj}
Mh
j=1 of Sh. If
uh(x, t) =
Mh∑
j=1
αj (t)Φj (x),
then (13.2) is equivalent to
Bα′′(t) + Aα(t) = b(t), for t > 0,
where the elements of B, A, and b are bkj = (Φj , Φk), akj = a(Φj , Φk), and
bk = (f, Φk), respectively. The initial conditions are
α(0) = β, α′(0) = γ,
where
vh =
Mh∑
j=1
βj Φj , wh =
Mh∑
j=1
γj Φj .
We begin by showing a discrete version of the energy conservation result
in Theorem 11.2.
Lemma 13.1. Let uh be the solution of (13.2) with f = 0. Then
∥uh,t(t)∥
2 + |uh(t)|
2
1 = ∥wh∥
2 + |vh|
2
1, for t ≥ 0.
Proof. Choosing χ = uh,t in (13.2) we have
1
2
d
dt
(
∥uh,t∥
2 + |uh|
2
1
)
= 0,
from which the result immediately follows. ⊓.
We now show the following error estimate, where Rh denotes the elliptic
projection defined in (5.49).
Theorem 13.1. Let uh and u be the solutions of (13.2) and (13.1). Then
we have, for t ≥ 0,
∥uh,t(t) − ut(t)∥ ≤ C
(
|vh − Rhv|1 + ∥wh − Rhw∥
)
+ Ch2
(
∥ut(t)∥2 +
∫ t
0
∥utt∥2 ds
)
,
∥uh(t) − u(t)∥ ≤ C
(
|vh − Rhv|1 + ∥wh − Rhw∥
)
+ Ch2
(
∥u(t)∥2 +
∫ t
0
∥utt∥2 ds
)
,
|uh(t) − u(t)|1 ≤ C
(
|vh − Rhv|1 + ∥wh − Rhw∥
)
+ Ch
(
∥u(t)∥2 +
∫ t
0
∥utt∥1 ds
)
.

13.1 The Wave Equation 203
Proof. Writing as usual
uh − u = (uh − Rhu) + (Rhu − u) = θ + ρ,
we may bound ρ and ρt as in the proof of Theorem 10.1 by
(13.3) ∥ρ(t)∥ + h|ρ(t)|1 ≤ Ch
2
∥u(t)∥2, ∥ρt(t)∥ ≤ Ch
2
∥ut(t)∥2.
For θ(t) we have, after a calculation analogous to that in (10.14),
(13.4) (θtt, χ) + a(θ, χ) = −(ρtt, χ), ∀χ ∈ Sh, for t > 0.
Imitating the proof of Lemma 13.1, we choose χ = θt:
1
2
d
dt
(
∥θt∥
2 + |θ|21
)
≤ ∥ρtt∥ ∥θt∥.
After integration in t we obtain
∥θt(t)∥
2 + |θ(t)|21 ≤ ∥θt(0)∥
2 + |θ(0)|21 + 2
∫ t
0
∥ρtt∥ ∥θt∥ ds
≤ ∥θt(0)∥
2 + |θ(0)|21 + 2
∫ t
0
∥ρtt∥ ds max
s∈[0,t]
∥θt∥
≤ ∥θt(0)∥
2 + |θ(0)|21 + 2
(∫ T
0
∥ρtt∥ ds
)2
+ 1
2
(
max
s∈[0,T ]
∥θt∥
)2
,
for t ∈ [0, T ]. This implies
1
2
(
max
s∈[0,T ]
∥θt∥
)2
≤ ∥θt(0)∥
2 + |θ(0)|21 + 2
(∫ T
0
∥ρtt∥ ds
)2
and hence
∥θt(t)∥
2 + |θ(t)|21 ≤ 2∥θt(0)∥
2 + 2|θ(0)|21 + 4
(∫ T
0
∥ρtt∥ ds
)2
,
for t ∈ [0, T ]. In particular this holds with t = T where T is arbitrary. Using
also bounds for ρtt similar to (13.3), we obtain
∥θt(t)∥ + ∥θ(t)∥ ≤ C
(
∥θt(t)∥ + |θ(t)|1
)
≤ C
(
∥wh − Rhw∥ + |vh − Rhv|1
)
+ Ch2
∫ t
0
∥utt∥2 ds,
and
|θ(t)|1 ≤ C
(
∥wh − Rhw∥ + |vh − Rhv|1
)
+ Ch
∫ t
0
∥utt∥1 ds.
Together with the bounds in (13.3) this completes the proof. ⊓.

204 13 The Finite Element Method for Hyperbolic Equations
We remark that the choices vh = Rhv and wh = Rhw in Theorem 13.1
give optimal order error estimates for all the three quantities considered, but
that other optimal choices of vh could cause a loss of one power of h, because
of the gradient in the first term on the right. This can be avoided by a more
refined argument. The regularity requirement on the exact solution can also
be reduced.
We shall now briefly discuss the discretization also in time, and let U n ∈
Sh denote the approximation at time tn = nk, where k is the time step. One
possible method is then to determine U n for n ≥ 2 by posing for n ≥ 1 the
equations
(13.5) (∂t∂̄tU
n, χ) + a( 1
4
(U n+1 + 2U n + U n−1), χ) = (f (tn), χ), ∀χ ∈ Sh,
where U 0 and U 1 are given approximations of u(0) = v and u(t1), respec-
tively. The choice of the average in the second term is motivated by a combi-
nation of stability and accuracy considerations. As regards stability we show
the following fully discrete analogue of the semidiscrete energy conservation
law of Lemma 13.1, where we define U n+1/2 = (U n + U n+1)/2.
Lemma 13.2. We have for the solution of (13.5), with f = 0,
∥∂tU
n
∥
2 + |U n+1/2|21 = ∥∂tU
0
∥
2 + |U 1/2|21, for n ≥ 0.
Proof. We apply (13.5) with
χ =
1
2k
(U n+1 − U n−1) =
1
2
(∂tU
n + ∂tU
n−1) =
1
k
(U n+1/2 − U n−1/2).
With this χ we have
(∂t∂̄tU
n, χ) =
1
2k
(∂tU
n
− ∂tU
n−1, ∂tU
n + ∂tU
n−1) =
1
2
∂̄t∥∂tU
n
∥
2
and
a( 1
4
U n+1 + 1
2
U n + 1
4
U n−1, χ) =
1
2k
a(U n+1/2 + U n−1/2, U n+1/2 − U n−1/2)
=
1
2
∂̄t|U
n+1/2
|
2
1.
Hence
∂̄t
(
∥∂tU
n
∥
2 + |U n+1/2|21
)
= 0,
from which the result follows. ⊓.
Using this stability result together with direct analogues of the arguments
in the proof of Theorem 13.1 one may show the following, where we use our
usual notation θn = U n − Rhu(tn). We leave the details to Problem 13.4.

13.2 First Order Hyperbolic Equations 205
Theorem 13.2. Let U n and u be the solutions of (13.5) and (13.1), and
assume that the initial values U 0 and U 1 are chosen in such a way that
∥∂tθ
0
∥ + |θ0|1 + |θ
1
|1 ≤ C(h
2 + k2).
Then, under the appropriate regularity conditions for u, we have, with C(u, t)
nondecreasing in t,
∥U n+1/2 − u(tn +
1
2
k)∥ + ∥∂tU
n
− ut(tn +
1
2
k)∥ ≤ C(u, tn)(h
2 + k2),
and
|U n+1/2 − u(tn +
1
2
k)|1 ≤ C(u, tn)(h + k
2), for n ≥ 0.
The conditions for the initial values may be satisfied by taking U 0 = Rhv
and U 1 = Rh(v + kw +
1
2
k2utt(0)), where utt(0) = ∆v + f (0).
Although Theorem 13.2 estimates the error at the points tn +
1
2
k it is clear
that optimal order approximations at the points tn may also be obtained,
since, e.g.,
∥
1
4
(U n+1 + 2U n + U n−1) − u(tn)∥
≤
1
2
∥U n+1/2 − u(tn +
1
2
k)∥ + 1
2
∥U n−1/2 − u(tn −
1
2
k)∥
+ ∥ 1
2
(u(tn +
1
2
k) + u(tn −
1
2
k)) − u(tn)∥ ≤ C(u, tn)(h
2 + k2).
13.2 First Order Hyperbolic Equations
We begin by considering the initial-boundary value problem, cf. Example
11.7,
(13.6)
ut + ux = f, in Ω = (0, 1), for t > 0,
u(0, t) = 0, for t > 0,
u(·, 0) = v, in Ω.
With 0 = x0 < x1 < · · · < xM = 1 and Kj = [xj−1, xj ], we shall seek an approximate solution in the space (13.7) S−h = { χ ∈ C(Ω̄) : χ linear in Kj , j = 1, . . . , M, χ(0) = 0 } . Note that we require the functions in S−h to vanish at x = 0, i.e., on the spatial part Γ−,x of the inflow boundary, but not at x = 1, which is part of the outflow boundary. The spatially discrete standard Galerkin method is then to find uh(t) ∈ S−h for t ≥ 0 such that (13.8) (uh,t + uh,x, χ) = (f, χ), ∀χ ∈ S − h , t > 0,
uh(0) = vh ≈ v.

206 13 The Finite Element Method for Hyperbolic Equations
In terms of the standard basis {Φj}
M
j=1 of hat functions this may be written
in the form
Bα′(t) + Aα(t) = f, for t > 0, with α(0) = γ,
where as usual B is the symmetric positive definite matrix with elements
bkj = (Φj , Φk), so that, in particular, the problem has a well defined so-
lution for t ≥ 0, but where the matrix A with elements akj = (Φ
′
j , Φk) =
−(Φ′k, Φj ) = −ajk is now skew-symmetric.
We begin to show the stability of this method, and choose χ = uh in
(13.8). This gives
1
2
d
dt
∥uh∥
2 + (uh,x, uh) = (f, uh) ≤ ∥f∥ ∥uh∥.
Here
(uh,x, uh) =
1
2
[
u2h
]1
0
= 1
2
uh(1)
2
≥ 0,
and thus
d
dt
∥uh∥ ≤ ∥f∥,
so that after integration
(13.9) ∥uh(t)∥ ≤ ∥vh∥ +
∫ t
0
∥f∥ ds, for t ≥ 0.
We now show an error estimate.
Theorem 13.3. Let uh and u be the solutions of (13.8) and (13.6). Then,
with vh suitably chosen, we have
∥uh(t) − u(t)∥ ≤ Ch
(
∥v∥1 +
∫ t
0
(
∥u∥2 + ∥ut∥1
)
ds
)
, for t ≥ 0.
Proof. With Ih the standard interpolation operator into Sh we write
uh − u = (uh − Ihu) + (Ihu − u) = θ + ρ.
Here by Theorem 5.5
∥ρ(t)∥ ≤ Ch∥u(t)∥1 ≤ Ch
(
∥v∥1 +
∫ t
0
∥ut∥1 ds
)
,
which is bounded as desired. By our definitions we have θ ∈ S−h and
(θt, χ) + (θx, χ) = −(ω, χ), ∀χ ∈ S
−
h , with ω = ρt + ρx.
From the stability estimate (13.9) and Theorem 5.5 we conclude, if vh = Ihv
so that θ(0) = 0,
∥θ(t)∥ ≤
∫ t
0
(
∥ρt∥ + ∥ρx∥
)
ds ≤ Ch
∫ t
0
∥ut∥1 ds + Ch
∫ t
0
∥u∥2 ds.
This completes the proof. ⊓.

13.2 First Order Hyperbolic Equations 207
We note that the error bound is not of optimal order O(h2) because the
bound for θ(t) contains the derivative of the interpolation error.
This analysis of the spatially semidiscrete problem may be carried over
to fully discrete methods. We exemplify this by the backward Euler method,
i.e., with our standard notation,
(13.10)
(∂̄tU
nχ) + (U nx , χ) = (f
n, χ), ∀χ ∈ S−h , n > 0,
U 0 = vh.
Now the stability bound is (Problem 13.5)
(13.11) ∥U n∥ ≤ ∥vh∥ + k
n∑
j=1
∥f j∥, for n ≥ 0,
and the error estimate reads as follows.
Theorem 13.4. Let U n and u be the solutions of (13.10) and (13.6). Then,
with vh suitably chosen, we have for n ≥ 0,
∥U n − u(tn)∥ ≤ Ch
(
∥v∥1 +
∫ tn
0
(∥u∥2 + ∥ut∥1) ds
)
+ Ck
∫ tn
0
∥utt∥ ds.
Proof. This time θn = U n − Ihu
n satisfies, with un = u(tn),
(∂̄tθ
n, χ) + (θnx , χ) = −(ω
n, χ), ∀χ ∈ S−h ,
where ωn = ∂̄tρ
n + ρnx + (u
n
t − ∂̄tu
n).
The only essentially new term in ωn is the last one, which is bounded by
∥unt − ∂̄tu
n
∥ =
∥
∥
∥
∫ tn
tn−1
(s − tn−1) utt(s) ds
∥
∥
∥ ≤ k
∫ tn
tn−1
∥utt∥ ds.
Using the stability estimate (13.11) completes the proof. ⊓.
In order to proceed further with finite element methods for equations of
first order we temporarily abandon the evolution aspect and consider the
two-dimensional problem, which we discussed in Sect. 11.3,
(13.12)
a · ∇u + a0u = f, in Ω,
u = g, on Γ−,
where, for brevity of presentation we assume that the velocity field a =
(a1, . . . , ad) and the coefficient a0 are constant with a0 > 0, and where we
recall that we have defined the inflow and outflow boundaries by
Γ− =
{
x ∈ Γ : a · n < 0 } , Γ+ = { x ∈ Γ : a · n > 0
}
.

208 13 The Finite Element Method for Hyperbolic Equations
We shall keep track of the dependence on the constant a0 in our estimates
below but assume that it is bounded above.
We now discretize this by means of a standard two-dimensional finite
element method. As in Sect. 5.2 we assume that Ω ⊂ R2 is a bounded
convex domain whose boundary Γ is a polygon, and we let Sh be a family
of spaces of piecewise linear finite element functions with respect to a family
of triangulations of Ω, without imposing any boundary conditions on the
functions in Sh. Thus, instead of Sh ⊂ H
1
0 , we now have Sh ⊂ H
1. We will use
the interpolation operator Ih defined in Sect. 5.3 and recall the interpolation
error estimates
(13.13) ∥Ihv − v∥ ≤ Ch
2
∥v∥2, |Ihv − v|1 ≤ Ch∥v∥2.
Finally, we assume that the triangulations are matched to the boundary so
that the inflow boundary is exactly a union of triangle edges and set
S−h =
{
χ ∈ Sh : χ = 0 on Γ−
}
.
We emphasize that the norms in (13.13) are taken over the two-dimensional
domain Ω.
The standard Galerkin finite element method for the present problem is
then to find uh ∈ Sh such that
(13.14)
(a · ∇uh, χ) + a0(uh, χ) = (f, χ), ∀χ ∈ S
−
h ,
uh = gh = Ihg, on Γ−,
where the inner products are now over the two-dimensional domain Ω.
Using Green’s formula we have the identity (recall that a is constant)
(13.15) (a · ∇v, v) = 1
2
(a · n v, v)Γ =
1
2
|v|2Γ+ −
1
2
|v|2Γ− ,
where we have introduced the weighted norms
|v|2Γ± = ±(a · n v, v)Γ± =
∫
Γ±
|a · n| v2 ds.
We consider now a solution wh ∈ Sh of (13.14), which satisfies the ho-
mogeneous boundary condition wh = 0 on Γ−. Since wh ∈ S
−
h we may then
choose χ = wh to obtain, in view of (13.15),
1
2
|wh|
2
Γ+
+ a0∥wh∥
2 = (f, wh).
For f = 0 this immediately shows that wh = 0 and hence the uniqueness
of the solution of (13.14), and therefore also the existence. We also easily
conclude the stability estimate
(13.16) |wh|
2
Γ+
+ a0∥wh∥
2
≤ C∥f∥2, with C = 1/a0.
We continue our discussion by proving the following simple error estimate.

13.2 First Order Hyperbolic Equations 209
Theorem 13.5. Let uh and u be the solutions of (13.14) and (13.12). Then
we have
∥uh − u∥ ≤ Ch∥u∥2.
Proof. We write uh − u = (uh − Ihu) + (Ihu − u) = θ + ρ. Then, in view of
(13.13), we have
(13.17) ∥ρ∥ + h∥ρ∥1 ≤ Ch
2
∥u∥2.
In order to estimate θ we note that θ ∈ S−h and, by (13.14) and (13.12),
(13.18) (a · ∇θ, χ) + a0(θ, χ) = −(a · ∇ρ + a0ρ, χ), ∀χ ∈ S
−
h .
Since θ ∈ S−h the stability estimate (13.16) together with (13.17) shows
|θ|2Γ+ + a0∥θ∥
2
≤ C(∥∇ρ∥2 + ∥ρ∥2) ≤ Ch2∥u∥22,
which completes the proof. ⊓.
We observe that, as in Theorems 13.3 and 13.4, the error estimate of
Theorem 13.5 is of non-optimal order O(h), as a result of the fact that the
gradient of the interpolation error occurs on the right hand side of (13.18),
and it is known that this error bound cannot be improved. Nevertheless,
this means that the standard Galerkin method works adequately when the
solution is smooth. However, solutions of (13.14) need not be smooth and
experience shows that the method then performs less well, and may, for ex-
ample, produce oscillations near layers where the solution changes rapidly.
In order to reduce such oscillations one may add artificial diffusion, as was
done to obtain the Friedrichs scheme from the unstable scheme (12.15). The
standard Galerkin method with artificial diffusion is to find uh ∈ Sh such
that
(13.19)
(a · ∇uh, χ) + a0(uh, χ) + h(∇uh, ∇χ) = (f, χ), ∀χ ∈ S
−
h .
uh = gh = Ihg, on Γ−.
This method is consistent with the elliptic equation a · ∇u + a0u − h∆u = f ,
and the error is therefore still expected to be O(h) for smooth solutions, see
Problem 13.6, and for non-smooth solutions the method has been observed
to smoothen discontinuities more than desirable.
More elaborate ways of adding diffusion have been developed. We now
describe one such method, the so-called streamline diffusion method, which
is to find uh ∈ Sh such that
(13.20)
(a · ∇uh + a0uh, χ + h a · ∇χ) = (f, χ + h a · ∇χ), ∀χ ∈ S
−
h .
uh = gh = Ihg, on Γ−.
We note that the exact solution of (13.12) satisfies

210 13 The Finite Element Method for Hyperbolic Equations
(13.21) (a · ∇u + a0u, χ + h a · ∇χ) = (f, χ + h a · ∇χ), ∀χ ∈ S
−
h ,
which means that (13.20) is consistent with (13.12). This method is an ex-
ample of a Petrov-Galerkin method in that we have chosen to multiply the
equation by test functions other than those in Sh.
The rest of this section will perhaps require a little harder work to get
through than has been the case with what we have presented earlier, but we
include this material because we think that it illustrates the difficulties in
applying the finite element method to first order hyperbolic equations.
We begin by discussing the stability and restrict ourselves again to a
solution wh ∈ S
−
h of (13.20), thus vanishing on Γ−. We then choose χ = wh,
and use (13.15) and ab ≤ a2 + 1
4
b2 to obtain
(13.22) 1
2
(1 + ha0)|wh|
2
Γ+
+ a0∥wh∥
2 + h ∥a·∇wh∥
2 = (f, wh) + h(f, a·∇wh).
As before this implies uniqueness and existence of solutions to (13.20) by
setting f = 0. Using the Cauchy-Schwarz inequality in the obvious way and
ha0 > 0 this yields the stability estimate
(13.23) |wh|
2
Γ+
+ a0∥wh∥
2 + h∥a · ∇wh∥
2
≤ (a−10 + h)∥f∥
2.
Note the extra stability given by the presence of the term h∥a · ∇wh∥
2. We
may interpret this by saying that the method adds artificial diffusion, but
only along the characteristic curves (streamlines).
For the error analysis we shall also need a somewhat stronger stability
estimate for the case that f has the form f = a · ∇F , which reads
(13.24) |wh|
2
Γ+
+ a0∥wh∥
2 + h∥a · ∇wh∥
2
≤ C(h∥F ∥21 + h
−1
∥F ∥2),
where C is independent of a0. Starting again with (13.22) we have now
h|(f, a · ∇wh)| = h|(a · ∇F, a · ∇wh)| ≤
1
4
h∥a · ∇wh∥
2 + Ch∥F ∥21.
Moreover, by Green’s formula,
(f, wh) = (a · ∇F, wh) = (a · n F, wh)Γ+ − (F, a · ∇wh),
and hence
|(f, wh)| ≤ C|F |
2
Γ+
+ 1
4
|wh|
2
Γ+
+ h−1∥F ∥2 + 1
4
h∥a · ∇wh∥
2.
Using the trace inequality
(13.25) |F |2Γ+ ≤ C∥F ∥ ∥F ∥1 ≤ Ch∥F ∥
2
1 + Ch
−1
∥F ∥2,
cf. Problem A.16, the proof of (13.24) is completed as in (13.22).
We are now ready for the following error estimate which shows an im-
provement of half a power of h compared to the standard Galerkin method.
We also remark that the error in the flux is of optimal order, ∥a·∇e∥ = O(h).

13.2 First Order Hyperbolic Equations 211
Theorem 13.6. Let uh and u be the solutions of (13.20) and (13.12). Then
we have for e = uh − u,
|e|Γ+ + a
1/2
0 ∥e∥ + h
1/2
∥a · ∇e∥ ≤ Ch3/2∥u∥2.
Proof. We write again uh − u = (uh − Ihu) + (Ihu − u) = θ + ρ and have
using (13.17) and (13.25)
|ρ|2Γ+ + a0∥ρ∥
2 + h∥a · ∇ρ∥2 ≤ C
(
h∥ρ∥21 + h
−1
∥ρ∥2
)
≤ Ch3∥u∥22.
This time we have by (13.20) and (13.21),
(a · ∇θ + a0θ, χ + h a · ∇χ) = −(a · ∇ρ + a0ρ, χ + h a · ∇χ), ∀χ ∈ S
−
h .
Choosing χ = θ ∈ S−h we conclude from (13.23) with f = a0ρ and (13.24)
with F = ρ, together with (13.17), with the last C depending only on an
upper bound for a0,
|θ|2Γ+ + a0∥θ∥
2 + h∥a · ∇θ∥2 ≤ C
(
a0∥ρ∥
2 + h∥ρ∥21 + h
−1
∥ρ∥2
)
≤ Ch3∥u∥22,
which completes the proof. ⊓.
The error estimate of the previous theorem thus shows that streamline
diffusion performs slightly better than the standard Galerkin method for
smooth solutions, but the main reason for its use is that it performs better
for non-smooth solutions, due to the fact that artificial diffusion is added
only in the characteristic direction so that internal layers are not smeared
out, while the added diffusion removes oscillations near boundary layers. We
shall not go into the details.
The result of Theorem 13.6 is valid also when a0 = 0. In this case the
problem (13.12) still admits a unique solution, since the uniqueness, and
hence the existence, again follows directly from (13.22) with f = 0. We have
assumed a0 > 0 in order to be able to bound the error to order O(h
3/2) in the
L2-norm. One may easily show the Poincaré type inequality ∥w∥ ≤ C∥a·∇w∥
for w = 0 on Γ−, and hence, in the absence of an estimate for ∥e∥ we then
have to be content with a O(h) error bound in L2-norm. We still have a
O(h3/2) error bound on Γ+, and in our next result we will solve the problem
in a sequence of domains in such a way that the bounds on the corresponding
Γ+ will result in a global O(h
3/2) error bound.
The above approach thus treats the first order hyperbolic problem as
a two-dimensional one, and, in particular, solves the discrete equations si-
multaneously for all the nodal values of the solution. Applied to the initial-
boundary value problem (13.6) the evolution aspect is therefore lost. We shall
now turn to a modification that retains the advantages of the streamline dif-
fusion method, but recovers the time stepping character. This will be done by
dividing the domain Ω ×R+ into strips parallel to the x-axis, and then using

212 13 The Finite Element Method for Hyperbolic Equations
approximating functions which are allowed to be discontinuous when passing
from one strip to the next. This method is referred to as the discontinuous
Galerkin method.
Considering thus the initial-boundary value problem (13.6), we use as
earlier the partition of Ω defined by 0 = x0 < x1 < · · · < xM = 1 and introduce now also a partition 0 = t0 < t1 < . . . of R+. We assume, for simplicity, that both partitions are quasi-uniform, and that the increments in space and time are of the same magnitude. Setting hj = xj − xj−1, kj = tj − tj−1, and h = max hj , k = max kj , this means that ch ≤ hj ≤ h, ck ≤ kj ≤ k, for all j, with c > 0, and that ch ≤ k ≤ Ch. These partitions
in space and time define a partition of Q = Ω × R+ into rectangles. These
could be further subdivided into triangles by means of diagonals with positive
slopes, say, to form a triangulation which would permit application of our
above discussion of the streamline diffusion method on any finite interval in
time. We shall use the undivided rectangles in our analysis, and define, with
Mn = (tn−1, tn),
Sh,k =
{
V (x, t) = αn(x)
t − tn−1
kn
+ βn(x)
tn − t
kn
,
for t ∈ Mn, with α
n, βn ∈ S−h
}
,
where S−h denotes the piecewise linear space defined in (13.7). Note that
V ∈ Sh,k may be discontinuous at tn and that V
+
n−1 = V (t
+
n−1) = β
n, V −n =
V (t−n ) = α
n, and Vt(t) = (α
n − βn)/kn for t ∈ Mn.
The discontinuous Galerkin method with streamline diffusion for the solu-
tion of (13.6) is to find U ∈ Sh,k such that U
−
0 = vh and then, for n = 1, 2, . . . ,
∫
Mn
(Ut + Ux, χ + h(χt + χx)) dt + (U
+
n−1, χ
+
n−1)
=
∫
Mn
(f, χ + h(χt + χx)) dt + (U
−
n−1, χ
+
n−1), ∀χ ∈ Sh,k,
(13.26)
where the inner products are over the one-dimensional interval Ω. We note
that if f and U −n−1 vanish, then we may choose χ = U to conclude easily
that U = 0 on Ω × Mn. Hence this equation can be solved for U
−
n and U
+
n−1,
if U −n−1 is given together with f on Ω × Mn, and the method is therefore a
time stepping procedure.
We remark that the local equation (13.26) is of the form (13.20) on the
domain Ω × Mn, with the boundary condition U = 0 on Γ−,x, but with
the boundary condition U +n−1 = U
−
n−1 on Γ−,t only weakly imposed, see
Problem 13.7.
By adding the equations in (13.26) and the initial condition (U −0 −
vh, χ
+
0 ) = 0 we can write the equations on Q in weak form as
Bn(U, χ) = Ln(vh, f ; χ), ∀χ ∈ Sh,k, for n ≥ 1,

13.2 First Order Hyperbolic Equations 213
where, with [v]j = v
+
j − v
−
j ,
Bn(v, w) =
n∑
j=1
∫
Mj
(vt + vx, w + h(wt + wx)) dt +
n−1∑
j=1
([v]j , w
+
j ) + (v
+
0 , w
+
0 )
and
(13.27) Ln(v, f ; w) = (v, w
+
0 ) +
n∑
j=1
∫
Mj
(f, w + h(wt + wx)) dt.
We note that since the exact solution is continuous in time, and thus the
jump terms vanish, it satisfies
(13.28) Bn(u, χ) = Ln(v, f ; χ), ∀χ ∈ Sh,k, for n ≥ 1.
By integration by parts we can write Bn(·, ·) as
Bn(v, w) =
n∑
j=1
∫
Mj
(
(v, −wt − wx) + h(vt + vx, wt + wx)
)
dt
+
n−1∑
j=1
(v−j , −[w]j ) + (v
−
n , w
−
n ) +
∫ tn
0
v(1, t)w(1, t) dt.
(13.29)
By adding the two forms of Bn(·, ·), using v
−
j = v
+
j −[v]j , we obtain for w = v
Bn(v, v) =
1
2
∥v−n ∥
2 + 1
2
n−1∑
j=1
∥[v]j∥
2 + h
n∑
j=1
∫
Mj
∥vt + vx∥
2 dt
+ 1
2
∥v+0 ∥
2 + 1
2
∫ tn
0
v(1, t)2 dt.
(13.30)
We now turn to the error analysis.
Theorem 13.7. Let U and u be the solutions of (13.26) and (13.6). Then,
for vh is suitably chosen, we have for e = U − u,
∥e−n ∥
2 +
n−1∑
j=1
∥[e]j∥
2 + h
n∑
j=1
∫
Mj
∥et + ex∥
2 dt
≤ Ch3
∫ tn
0
(
∥u∥22 + ∥ut∥
2
1 + ∥utt∥
2
)
dt, for n ≥ 0.
(13.31)
We remark that the first term on the left shows a O(h3/2) estimate for the
error to the left at tn. Since the jumps at the time levels are bounded in
the second term, the error to the right at tn, e
+
n = e
−
n + [e]n, is also of
order O(h3/2), and we may conclude that this holds everywhere on Mn, since
e(t) = k−1(t − tn−1)e
−
n + k
−1(tn − t)e
+
n−1 + (ū(t) − u(t)), where ū denotes the
linear interpolant of u so that the last term is O(h2).

214 13 The Finite Element Method for Hyperbolic Equations
Proof of Theorem 13.7. The proof proceeds in a way similar to that of The-
orem 13.6 and all terms that occur below have their counterparts there. Let
Ih denote the interpolation operator into S
−
h and Jk the piecewise linear
interpolation operator in time. We write
U − u = (U − ũ) + (ũ − u) = θ + ρ, where ũ = JkIhu.
Note that ũ(·, t) is continuous in time. In view of (13.30) it suffices to bound
Bn(e, e), and we note that
Bn(e, e) ≤ 2Bn(θ, θ) + 2Bn(ρ, ρ).
We begin by bounding the first term on the right. We find in view of (13.28)
and (13.26), for any χ ∈ Sh,k,
Bn(θ, χ) = Bn(U, χ) − Bn(ũ, χ) = Ln(vh, f ; χ) − Bn(ũ, χ)
= Ln(vh, f ; χ) +
(
Bn(u, χ) − Ln(v, f ; χ)
)
− Bn(ũ, χ)
= (vh − v, χ
+
0 ) − Bn(ρ, χ) = (e0, χ
+
0 ) − Bn(ρ, χ) = −Bn(ρ, χ),
where we have now chosen the initial value vh = Phv so that (e0, χ
+
0 ) = 0.
Setting χ = θ and using (13.29) for Bn(·, ·), we conclude
Bn(θ, θ) = |Bn(ρ, θ)| ≤
n∑
j=1
∫
Mj
(
∥ρ∥ + h∥ρt + ρx∥
)
∥θt + θx∥ dt
+
n−1∑
j=1
∥ρj∥ ∥[θ]j∥ + ∥ρn∥ ∥θ
−
n ∥ +
∫ tn
0
|ρ(1, t)| |θ(1, t)| dt
≤
1
2
Bn(θ, θ) + Ch
−1
∫ tn
0
∥ρ∥2 dt +
n∑
j=1
∥ρj∥
2 + CBn(ρ, ρ).
Here we noted that ρ is continuous in time so that ρ−n = ρn. To complete the
proof we kick back Bn(θ, θ) and then bound the last three terms appropri-
ately. Note first that, by (13.30),
Bn(ρ, ρ) =
1
2
∥ρn∥
2 + h
∫ tn
0
∥ρt + ρx∥
2 dt + 1
2
∥ρ0∥
2 + 1
2
∫ tn
0
ρ(1, t)2 dt.
We write
ρ = JkIhu − u = Jk(Ihu − u) + (Jku − u) = Jkη + ω.
Using the standard estimates for Jk, with ∥v∥
2
Mj
=
∫
Mj
∥v∥2 dt,
∥Jkv − v∥Mj + kj∥Dt(Jkv − v)∥Mj ≤ Ck
s
j ∥D
s
t v∥Mj , for s = 1, 2,

13.2 First Order Hyperbolic Equations 215
and since the partitions are quasi-uniform, with h and k of the same order of
magnitude, we get
n∑
j=1
∫
Mj
(
h−1∥ω∥2 + h∥ωt∥
2 + h∥ωx∥
2
)
dt
≤ C(h−1k4 + hk2)
∫ tn
0
∥utt∥
2 dt + Chk2
∫ tn
0
∥ut∥
2
1 dt
≤ Ch3
∫ tn
0
(
∥utt∥
2 + ∥ut∥
2
1
)
dt.
Next we note that ∥Jkη(t)∥ ≤ maxs∈Mj ∥η(s)∥ for t ∈ Mj , and use a trace
inequality from Problem A.12, to see that
∥η(t)∥2 ≤ Ck−1j
∫
Mj
∥η∥2 dt + Ckj
∫
Mj
∥ηt∥
2 dt
≤ Ch3
∫
Mj
(
∥u∥22 + ∥ut∥
2
1
)
dt, for t ∈ Mj ,
(13.32)
so that
h−1
∫ tn
0
∥Jkη∥
2 dt ≤ Ch3
∫ tn
0
(
∥u∥22 + ∥ut∥
2
1
)
dt.
In a similar way, using
∥(Jkη)t(t)∥ = k
−1
j ∥ηj − ηj−1∥ ≤ 2k
−1
j max
s∈Mj
∥η(s)∥, for t ∈ Mj ,
we get
h
n∑
j=1
∫
Mj
(
∥(Jkη)x∥
2 + ∥(Jkη)t∥
2
)
dt ≤ Ch3
∫ tn
0
(∥u∥22 + ∥ut∥
2
1) dt.
Further, from (13.32),
n∑
j=0
∥ρj∥
2 =
n∑
j=0
∥ηj∥
2
≤ Ch3
∫ tn
0
(
∥u∥22 + ∥ut∥
2
1
)
dt.
Finally, using again a trace inequality from Problem A.12, we get
∫ tn
0
|ρ(1, t)|2 dt =
∫ tn
0
|ω(1, t)|2 dt ≤ C
∫ tn
0
∥ω(·, t)∥ ∥ω(·, t)∥1 dt
≤ Ck3
(∫ tn
0
∥utt∥
2 dt
∫ tn
0
∥ut∥
2
1 dt
)1/2
≤ Ch3
∫
Mj
(
∥utt∥
2 + ∥ut∥
2
1
)
dt.
This completes the proof. ⊓.

216 13 The Finite Element Method for Hyperbolic Equations
13.3 Problems
Problem 13.1. Consider the initial-boundary value problem
utt = uxx, x ∈ (0, 1), t > 0,
u(0, t) = u(1, t) = 0, t > 0,
u(x, 0) = v(x), ut(x, 0) = w(x), x ∈ (0, 1).
For the numerical solution by the Galerkin finite element method, consider
the piecewise linear continuous functions based on the partition of [0, 1] into
M intervals of equal lengths h = 1/M . Find the matrix forms of the semidis-
crete and completely discrete methods analogous to those described above in
(13.2) and (13.5).
Problem 13.2. (Computer exercise.) Solve the initial-boundary value prob-
lem in Problem 13.1 with v(x) = 0, w(x) = sin(2πx) using M = 10, 20 and
the time stepping method in (13.5) with k = 1/10, 1/20, and compare with
the exact solution given in Sect. 11.2 at time t = 3/4.
Problem 13.3. Write the wave equation (13.1) as a system of two equations
of first order in time by setting w1 = u, w2 = ut. Discretize this system by
means of the standard finite element method in the spatial variables and by
means of the Crank-Nicolson method in the time variable. Show, by elimina-
tion of W n2 , that the resulting scheme is essentially the same as (13.5). Prove
stability in the case f = 0. Compare with Lemma 13.2. Hint: Multiply the
system by (W
n− 1
2
1 , −∆hW
n− 1
2
2 ).
Problem 13.4. Prove Theorem 13.2.
Problem 13.5. Show the stability bound (13.11).
Problem 13.6. Prove stability and error estimates for the standard Galerkin
method with artificial diffusion (13.19).
Problem 13.7. (Weakly imposed boundary condition.) The boundary con-
dition u = g is imposed strongly in (13.14) and (13.20). It is also possible
to impose the boundary condition weakly in the standard Galerkin method:
Find uh ∈ Sh such that
(a · ∇uh, χ) + (a0uh, χ) − (a · n uh, χ)Γ− = (f, χ) − (a · n g, χ)Γ− , ∀χ ∈ Sh,
and in its streamline diffusion modification: Find uh ∈ Sh such that
(a · ∇uh, χ + ha · ∇χ) + (a0uh, χ + ha · ∇χ) − (a · n uh, χ)Γ−
= (f, χ + ha · ∇χ) − (a · n g, χ)Γ− , ∀χ ∈ Sh.
Prove stability and error estimates for these methods.

14 Some Other Classes of Numerical Methods
Numerical methods other than finite difference and finite element methods,
but often closely related to these, have been developed and are also of inter-
est. In this chapter we review briefly four such classes of methods, namely
collocation methods, spectral methods, finite volume methods, and boundary
element methods.
14.1 Collocation methods
In a collocation method one seeks an approximate solution of a differential
equation in a finite dimensional space of sufficiently regular functions by
requiring that the equation is satisfied exactly at a finite number of points.
We describe one such method for the parabolic model problem
ut = uxx in Ω = (0, 1), for t > 0,
u(0, t) = u(1, t) = 0 for t > 0,
u(·, 0) = v in Ω.
Setting h = 1/M, xj = jh for 0 ≤ j ≤ M , and Kj = [xj−1, xj ], we introduce
the piecewise polynomial space
Sh =
{
v ∈ C1(Ω̄) : v
∣
∣
Kj
∈ Πr−1, v(0) = v(1) = 0
}
, with r ≥ 4.
Letting ξi, i = 1, . . . , r − 2, be the Gauss points in (0, 1), i.e., the zeros of
the Legendre polynomial P̃r−2(x) = Pr−2(2x − 1), re-scaled from the interval
(−1, 1) to (0, 1), we define the collocation points xj,i = xj−1 + hξi in Kj , and
pose the spatially semidiscrete problem to find uh(·, t) ∈ Sh for t > 0 such
that
(14.1) uh,t(xj,i, t) = uh,xx(xj,i, t), for 1 ≤ j ≤ M, 1 ≤ i ≤ r − 2, t > 0,
with uh(·, 0) = vh ∈ Sh an approximation of v. This method may be consid-
ered as a Galerkin method using a discrete inner product based on the Gauss
quadrature rule. In fact, letting ωi be the weights in the Gauss formula
r−2∑
i=1
ωi ϕ(ξi) ≈
∫ 1
0
ϕ(x) dx,

218 14 Some Other Classes of Numerical Methods
which is exact for polynomials of degree at most 2r − 5, we set
(14.2) (ψ, χ)h = h
M∑
j=1
r−2∑
i=1
ωi ψ(xj,i) χ(xj,i) ≈ (ψ, χ),
and we may then write (14.1) as
(uh,t, χ)h − (uh,xx, χ)h = 0, ∀χ ∈ Sh, t > 0.
For vh appropriately chosen one may show the global error estimate
∥uh(t) − u(t)∥C ≤ Ch
r
{
max
s≤t
∥u(s)∥r+2 +
(∫ t
0
∥ut(s)∥
2
r+2 ds
)1/2}
.
Further, for r > 4, and with a more refined choice of initial approximation
vh, superconvergence takes place at the nodes, so that
|uh(xj , t) − u(xj , t)| ≤ CT h
2r−4 sup
s≤t
∑
p+2q≤2r−1
∥u(q)(s)∥p, for t ≤ T.
We note the more stringent regularity requirements than for the finite differ-
ence and finite element methods discussed in Chaps. 9–10. These results carry
over to fully discrete schemes using both finite difference approximations and
collocation in time.
14.2 Spectral Methods
Spectral methods are in many ways similar to finite element and collocation
methods. The main difference is in the choice of finite dimensional approxi-
mating spaces.
Consider the initial value problem
(14.3)
ut − uxx = f in Ω = (0, 1), for t > 0,
u(0, t) = u(1, t) = 0 for t > 0,
u(·, 0) = v in Ω.
Let now {ϕj}
∞
j=1 be a sequence of linearly independent functions in H
2 ∩H10 ,
which span L2, and set SN = span{ϕj}
N
j=1. Using Galerkin’s method we
define a spatially semidiscrete approximation uN = uN (t) ∈ SN of (14.3) by
(14.4) (uN,t, χ) − (uN,xx, χ) = (f, χ), ∀χ ∈ SN , t > 0,
with uN (0) = vN ∈ SN suitably chosen. Introducing the orthogonal projec-
tion PN : L2 → SN , we may write (14.4) as
uN,t + AN uN = PN f, for t > 0, where AN = PN APN , A = ∂
2/∂x2.

14.3 Finite Volume Methods 219
Here (AN χ, χ) = (APN χ, PN χ) ≥ 0. With uN (x, t) =
∑N
j=1 αj (t)ϕj (x), this
equation may be written Bα′(t) + Aα(t) = b(t) for t > 0, where the elements
of the N by N matrices A and B are (Aϕi, ϕj ) and (ϕi, ϕj ), respectively. It
is easy to see that B is positive definite.
We note that the error eN = uN − u satisfies
eN,t + AN eN = (PN − I)f − (AN − A)u for t > 0, with eN (0) = vN − v,
and hence, since the corresponding solution operator EN (t) = e
−AN t is easily
seen to be bounded by 1 in L2 operator norm,
(14.5) ∥uN (t) − u(t)∥ ≤ ∥vN − v∥ +
∫ t
0
(
∥(PN − I)f∥ + ∥(AN − A)u∥
)
ds.
It follows that the error is small with vN − v, (PN − I)f , and (AN − A)u.
As a simple example, let the ϕj (x) = c sin(jπx) be the normalized
eigenfunctions of A, with homogeneous Dirichlet boundary conditions. Then
B = I, A is positive definite and PN is simply the truncation of the Fourier
series, PN v =
∑N
j=1(v, ϕj )ϕj , so that AN v =
∑N
j=1(jπ)
2(v, ϕj )ϕj = PN Av.
Thus, if vN = PN v and if the Fourier series of v, f, and uxx converge, then
the error is small. In particular, the convergence is of order O(N −r) for any
r, provided that the solution is sufficiently regular.
Another way to define a semidiscrete numerical method employing the
space SN of our example is to make SN a Hilbert space with the inner product
(v, w)N = h
∑N−1
j=0 v(xj )w(xj ), where xj = j/(N − 1). This gives rise to
a projection PN defined by PN u(xj ) = u(xj ), j = 0, . . . , N − 1, and the
semidiscrete equation (14.4) now becomes the collocation equation
uN,t(xj , t) − uN,xx(xj , t) = f (xj , t), for j = 0, . . . , N − 1, t > 0.
This is also referred to as a pseudospectral method and the error estimate
(14.5) is valid in the discrete norm corresponding to (·, ·)N .
Spectral and pseudospectral methods using the above sinusoidal basis
functions are particularly useful for periodic problems. For initial-boundary
value problems for hyperbolic equations basis functions related to Chebyshev
and Legendre polynomials are sometimes useful, e.g., in connection with fluid
dynamics calculations.
14.3 Finite Volume Methods
We illustrate the use of the finite volume methods in the case of the model
problem
(14.6) −∆u = f in Ω, with u = 0 on Γ,

220 14 Some Other Classes of Numerical Methods
where Ω is a convex polygonal domain in R2 with boundary Γ . The basis for
this approach is the observation that, by Green’s formula, for any V ⊂ Ω we
have
(14.7)
∫
∂V
∂u
∂n
ds =
∫
V
f dx.
We begin by describing the cell centered finite volume difference method.
Let Th = {Kj} be a triangulation of Ω of the type considered in Sect. 5 in
which all angles of the Kj are < π/2, and consider (14.7) with V = Kj ∈ Th. Then ∂V = ∂Kj is the union of the edges γji common with three other triangles Ki, and we want to approximate ∂u/∂n on each of these edges. With Qj the center of the circumscribed circle of Kj (which then lies in the interior of Kj ), the vector Qj Qj is orthogonal to γji, and ∂u/∂n in (14.7) may be approximated by the difference quotient (U (Qi) − U (Qj ))/|QiQj|. Using the boundary values in (14.6) for the Qj associated with the boundary triangles, this produces a finite difference scheme on the nonuniform mesh {Qj}. Writing the discrete problem in matrix form as AU = b, one may show that the matrix A is symmetric positive definite and diagonally dominant. When the Th are quasi-uniform one may show the error estimate ∥U − u∥1,h ≤ Ch∥u∥2 in a certain discrete H1-norm. An alternative approach is the following vertex centered method, also re- ferred to as the finite volume element method : Let Sh ⊂ H 1 0 be the piecewise linear finite element space defined by Th. For K ∈ Th the straight lines con- necting a vertex with the midpoint of the opposite edge intersect at the barycenter of K and divide K into six triangles. Let Bj,K be the union of the two of these which have Pj as a vertex. For each interior vertex Pj we consider the union Bj of the corresponding Bj,K , and let S̄h denote the as- sociated piecewise constant functions. Using (14.7) for each of the Bj we are lead to the Petrov-Galerkin method to find uh ∈ Sh such that (14.8) ā(uh, ψ) := ∑ j ψj ∫ ∂Bj ∂uh ∂n ds = (f, ψ) ∀ψ ∈ S̄h, which may also be thought of as a finite difference scheme on the irregular mesh {Pj}. The Bj are referred to as control volumes. Associating with χ ∈ Sh the function χ̄ ∈ S̄h, which agrees with χ at the vertices of Th, one may show that (cf. Problem 14.3) (14.9) ā(ψ, χ̄) = a(ψ, χ), ∀ψ, χ ∈ Sh, so that (14.8) may be written a(uh, χ) = (f, χ̄), ∀χ ∈ Sh. 14.4 Boundary Element Methods 221 (This does not hold exactly for elliptic operators with variable coefficients.) It may be shown that the standard error estimate ∥uh − u∥1 ≤ Ch∥u∥2 holds for this method, and also, under slightly more stringent regularity as- sumptions, that ∥uh − u∥ = O(h 2). Finite volume methods are useful for operators in divergence form, par- ticularly for time dependent conservation laws. 14.4 Boundary Element Methods In a boundary integral method a boundary value problem for a homogeneous partial differential equation in a domain Ω with the solution u given on the boundary Γ is reformulated as an integral equation over Γ . This equation may then be used as a basis for numerical approximation. We shall illustrate this approach for the model problem (14.10) ∆u = 0 in Ω ⊂ R2, with u = g on Γ, where we assume Γ smooth. To pose the boundary integral equation, let U (x) = −(2π)−1 log |x| be the fundamental solution of the Laplacian in R2, see Theorem 3.5. For any u with ∆u = 0 on Γ we have by Green’s formula (14.11) u(x) = ∫ Γ U (x − y) ∂u ∂ny (y) dsy − ∫ Γ ∂U ∂ny (x − y)u(y) dsy, x ∈ Ω. With x on Γ the integrals on the right define the single and double layer po- tentials V ∂u/∂n and W u. We note that although ∇U (x−y) has a singularity of order O(|x − y|−1), the kernel (∂U/∂ny)(x − y) is bounded for x, y ∈ Γ , so that the operator W is well defined. For x ∈ Ω approaching Γ the two integrals tend to V ∂u/∂n and 1 2 u + W u, respectively, so that (14.11) yields 1 2 u = V ∂u/∂n − W u on Γ. With u = g on Γ this is a Fredholm integral equation of the first kind for the determination of ∂u/∂n on Γ , which inserted into (14.11) together with u = g on Γ gives the solution of (14.10). Instead of this direct method one may use the indirect method of assuming that the solution of (14.11) may be represented as a potential of a function on Γ , so that u(x) = ∫ Γ Φ(x − y)v(y) dsy or u(x) = ∫ Γ ∂Φ ∂ny (x − y)w(y) dsy, x ∈ Ω. With V and W as above, if such functions v and w exist, they satisfy the first and second kind Fredholm integral equations 222 14 Some Other Classes of Numerical Methods (14.12) V v = g and 1 2 w + W w = g on Γ. Writing Hs = Hs(Γ ), V and W are so-called pseudodifferential operators of order −1, i.e., bounded linear operators Hs → Hs+1, in particular compact on Hs. The first kind equation is uniquely solvable provided a certain measure, the transfinite diameter δΓ of Γ , is such that δΓ ̸= 1, and the second kind equation in (14.12) always has a unique solution. Similar reformulations may be used also for Neumann boundary conditions, for a large number of other problems involving elliptic type equations, and for exterior problems; in fact, this approach to the numerical solution is particularly useful in the latter case. In the Boundary Element Method (BEM) one determines the approximate solution in a piecewise polynomial finite element type space of a boundary integral formulation such as the above, using the Galerkin or the collocation method. For the second kind equation in (14.12), using Galerkin’s method and a finite dimensional subspace Sh of L2(Γ ), we determine the discrete approxi- mation wh ∈ Sh to w from 1 2 ⟨wh, χ⟩ + ⟨W wh, χ⟩ = ⟨g, χ⟩, ∀χ ∈ Sh, where ⟨·, ·⟩ = (·, ·)L2(Γ ). Writing | · |s for the norm in H s(Γ ), one has |wh − w|0 ≤ Cr(w)h r if Sh is accurate of order O(hr), and by a duality argument one may show the superconvergent order negative norm estimate |wh − w|−r ≤ Cr(w)h 2r; using an iteration argument this may be used to define an approximate solution w̃h with |w̃h − w|0 = O(h 2r). Consider for example the numerical solution of the first kind equation in (14.12) in the finite dimensional space Sh of periodic smoothest splines of order r, i.e., Sh ⊂ C r−2 consists of piecewise polynomials in Πr−1. Our discrete problem is then to find vh ∈ Sh such that ⟨V vh, χ⟩ = ⟨g, χ⟩, ∀χ ∈ Sh. It can be shown that the bilinear form ⟨V v, w⟩ associated with V is sym- metric, bounded, and coercive with respect to the norm | · |−1/2 in a certain Sobolev space H−1/2(Γ ), so that ⟨V v, w⟩ = ⟨v, V w⟩ ≤ C|v|−1/2|w|−1/2 and ⟨V v, v⟩ ≥ c|v| 2 −1/2, with c > 0.
One may then show that
|vh − v|−1/2 ≤ C inf
χ∈Sh
|χ − v|−1/2 ≤ Ch
r+1/2
|v|r,
and a duality argument implies |vh − v|−r−1 ≤ Ch
2r+1|v|r, where we use
the norm in H−r−1(Γ ). For x an interior point of Ω we therefore find for
uh = V vh that |uh(x) − u(x)| ≤ Cx|vh − v|−r−1 ≤ Ch
2r+1, since Φ(x − y) is
smooth when y ̸= x.

14.5 Problems 223
Expressed in terms of a basis {φj} of Sh this problem may be written
in matrix form as Aα = g̃, where A is symmetric positive definite. However,
although the dimension of A has been reduced by the reduction of the original
two-dimensional problem to a one-dimensional one, in contrast to the finite
element method for a differential equation problem, the matrix A is now not
sparse. We also note that the elements ⟨V Φi, Φj⟩ require two integrations,
one in forming V Φi and one in forming the inner product.
In order to reduce this work one may apply the collocation method and
determine vh from V vh(x(sj )) = g(x(sj )) at Mh quadrature points sj in [0, l],
where x = x(s) is a parametrization of Γ and Mh = dim(Sh).
In the vast literature on the numerical boundary integral methods much
attention has been paid to the complications arising when our above regular-
ity assumptions fail to be satisfied, such as for domains with corners in which
case V and W are not compact.
14.5 Problems
Problem 14.1. Let r = 4 and let (·, ·)h be defined by the corresponding case
of (14.2).
(a) Show that ∥χ∥h := (χ, χ)
1/2
h is a norm on Sh.
(b) Show that
−(χ′′, χ)h ≥ −(χ
′′, χ) = ∥χ′∥2, for χ ∈ Sh.
(c) Show the stability of the solution of (14.1) with respect to ∥ · ∥h.
Hint for (b): Let P̃2(x) = P2(2x − 1) = x
2 − x + 1
6
be the Legendre
polynomial corresponding to (0, 1) with zeros ξ1,2 =
1
2
±
√
3
6
. Recall that Gauss
quadrature with two Gauss points is exact for cubic polynomials. Restrict the
consideration to one interval (0, 1) and let χ ∈ Π3 with coefficient 1 for x
3.
Then χ′′χ − 6P̃ 22 ∈ Π3 and hence, since P̃ (ξi) = 0, i = 1, 2,
−
1
2
2∑
i=1
χ′′(ξi)χ(ξi) = −
∫ 1
0
χ′′χ dx + 6
∫ 1
0
P̃ 22 dx ≥ −
∫ 1
0
χ′′χ dx.
Problem 14.2. Consider the first order initial value problem with periodic
boundary conditions
ut + ux = 0, in Ω = (−π, π), for t > 0,
u(−π, t) = u(π, t), for t > 0,
u(·, 0) = v, in Ω.
Formulate the spectral method based on

224 14 Some Other Classes of Numerical Methods
SN =
{
1, cos x, sin x, cos 2x, sin 2x, . . . , cos N x, sin N x
}
.
Let A = ∂/∂x, determine AN and show that A
∗
N = −AN , where ∗ denotes the
adjoint. Show also that ∥uN (t)∥ ≤ ∥v∥ = ∥v∥L2(Ω), and hence that ∥EN (t)∥ =
1, where EN (t) = e
−tAN .
Problem 14.3. Show (14.9). Hint: Write Ω as unions of the Bj and of the
K, and write these in turn as unions of the sets Bj,K . Note that
∫
e
χ̄ ds =
∫
e
χ ds, for χ ∈ Sh,
for any edge e of the triangulation Th.

A Some Tools from Mathematical Analysis
In this appendix we give a short survey of results, essentially without proofs,
from mathematical, particularly functional, analysis which are needed in our
treatment of partial differential equations. We begin in Sect. A.1 with a simple
account of abstract linear spaces with emphasis on Hilbert space, including
the Riesz representation theorem and its generalization to bilinear forms of
Lax and Milgram. We continue in Sect. A.2 with function spaces, where after
a discussion of the spaces Ck, integrability, and the Lp-spaces, we turn to
L2-based Sobolev spaces, with the trace theorem and Poincaré’s inequality.
The final Sect. A.3 is concerned with the Fourier transform.
A.1 Abstract Linear Spaces
Let V be a linear space (or vector space) with real scalars, i.e., a set such
that if u, v ∈ V and λ, µ ∈ R, then λu + µv ∈ V . A linear functional (or
linear form) L on V is a function L : V → R such that
L(λu + µv) = λL(u) + µL(v), ∀u, v ∈ V, λ, µ ∈ R.
A bilinear form a(·, ·) on V is a function a : V × V → R, which is linear in
each argument separately, i.e., such that, for all u, v, w ∈ V and λ, µ ∈ R,
a(λu + µv, w) = λa(u, w) + µa(v, w),
a(w, λu + µv) = λa(w, u) + µa(w, v).
The bilinear form a(·, ·) is said to be symmetric if
a(w, v) = a(v, w), ∀v, w ∈ V,
and positive definite if
a(v, v) > 0, ∀v ∈ V, v ̸= 0.
A positive definite, symmetric, bilinear form on V is also called an inner
product (or scalar product) on V . A linear space V with an inner product is
called an inner product space.

226 A Some Tools from Mathematical Analysis
If V is an inner product space and (·, ·) is an inner product on V , then
we define the corresponding norm by
(A.1) ∥v∥ = (v, v)1/2, for v ∈ V.
We recall the Cauchy-Schwarz inequality,
(A.2) |(w, v)| ≤ ∥w∥ ∥v∥, ∀v, w ∈ V,
with equality if and only if w = λv or v = λw for some λ ∈ R, and the
triangle inequality,
(A.3) ∥w + v∥ ≤ ∥w∥ + ∥v∥, ∀v, w ∈ V.
Two elements v, w ∈ V for which (v, w) = 0 are said to be orthogonal.
An infinite sequence {vi}
∞
i=1 in V is said to converge to v ∈ V , also written
vi → v as i → ∞ or v = limi→∞ vi, if
∥vi − v∥ → 0 as i → ∞.
The sequence {vi}
∞
i=1 is called a Cauchy sequence in V if
∥vi − vj∥ → 0 as i, j → ∞.
The inner product space V is said to be complete if every Cauchy sequence
in V is convergent, i.e., if every Cauchy sequence {vi}
∞
i=1 has a limit v =
lim vi ∈ V . A complete inner product space is called a Hilbert space.
When we want to emphasize that an inner product or a norm is associated
to a specific space V , we write (·, ·)V and ∥ · ∥V .
It is sometimes important to permit the scalars in a linear space V to be
complex numbers. Such a space is then an inner product space if there is a
functional (v, w) defined on V × V , which is linear in the first variable and
hermitian, i.e., (w, v) = (v, w). The norm is then again defined by (A.1) and
V is a complex Hilbert space if completeness holds with respect to this norm.
For brevity we generally consider the case of real-valued scalars in the sequel.
More generally, a norm in a linear space V is a function ∥ · ∥ : V → R+
such that
∥v∥ > 0, ∀v ∈ V, v ̸= 0,
∥λv∥ = |λ| ∥v∥, ∀λ ∈ R (or C), v ∈ V,
∥v + w∥ ≤ ∥v∥ + ∥w∥, ∀v, w ∈ V.
A function |·| is called a seminorm if these conditions hold with the exception
that the first one is replaced by |v| ≥ 0, ∀v ∈ V , i.e., if it is only positive
semidefinite, and thus can vanish for some v ̸= 0. A linear space with a norm
is called a normed linear space. As we have seen, an inner product space is
a normed linear space, but not all normed linear spaces are inner product
spaces. A complete normed space is called a Banach space.

A.1 Abstract Linear Spaces 227
Let V be a Hilbert space and let V0 ⊂ V be a linear subspace. Such a
subspace V0 is said to be closed if it contains all limits of sequences in V0,
i.e., if {vj}
∞
j=1 ⊂ V0 and vj → v as j → ∞ implies v ∈ V0. Such a V0 is itself
a Hilbert space, with the same inner product as V .
Let V0 be a closed subspace of V . Then any v ∈ V may be written uniquely
as v = v0 + w, where v0 ∈ V0 and w is orthogonal to V0. The element v0 may
be characterized as the unique element in V0 which is closest to v, i.e.,
(A.4) ∥v − v0∥ = min
u∈V0
∥v − u∥.
This is called the projection theorem and is a basic result in Hilbert space
theory. The element v0 is called the orthogonal projection of v onto V0 and is
also denoted PV0 v. One useful consequence of the projection theorem is that
if the closed linear subspace V0 is not equal to the whole space V , then it has
a normal vector, i.e., there is a nonzero vector w ∈ V which is orthogonal to
V0.
Two norms ∥ · ∥a and ∥ · ∥b are said to be equivalent in V if there are
positive constants c and C such that
(A.5) c∥v∥b ≤ ∥v∥a ≤ C∥v∥b, ∀v ∈ V.
Let V, W be two Hilbert spaces. A linear operator B : V → W is said to
be bounded, if there is a constant C such that
(A.6) ∥Bv∥W ≤ C∥v∥V , ∀v ∈ V.
The norm of a bounded linear operator B is
(A.7) ∥B∥ = sup
v∈V \{0}
∥Bv∥W
∥v∥V
.
Thus
∥Bv∥W ≤ ∥B∥ ∥v∥V , ∀v ∈ V,
and, by definition, ∥B∥ is the smallest constant C such that (A.6) holds.
Note that a bounded linear operator B : V → W is continuous. In fact,
if vj → v in V , then Bvj → Bv in W as j → ∞, because
∥Bvj − Bv∥W = ∥B(vj − v)∥W ≤ ∥B∥ ∥vj − v∥ → 0, as j → ∞.
One can show that, conversely, a continuous linear operator is bounded.
In the special case that W = R the definition of an operator reduces to
that of a linear functional. The set of all bounded linear functionals on V is
called the dual space of V , denoted V ∗. By (A.7) the norm in V ∗ is
(A.8) ∥L∥V ∗ = sup
v∈V \{0}
|L(v)|
∥v∥V
.

228 A Some Tools from Mathematical Analysis
Note that V ∗ is itself a linear space if we define (λL + µM )(v) = λL(v) +
µM (v) for L, M ∈ V ∗, λ, µ ∈ R. With the norm defined by (A.8), V ∗ is a
normed linear space, and one may show that V ∗ is complete, and thus itself
also a Banach space.
Similarly, we say that the bilinear form a(·, ·) on V is bounded if there is
a constant M such that
(A.9) |a(w, v)| ≤ M∥w∥ ∥v∥, ∀w, v ∈ V.
The next theorem states an important property of Hilbert spaces.
Theorem A.1. (Riesz’ representation theorem.) Let V be a Hilbert space
with scalar product (·, ·). For each bounded linear functional L on V there is
a unique u ∈ V such that
L(v) = (v, u), ∀v ∈ V.
Moreover,
(A.10) ∥L∥V ∗ = ∥u∥V .
Proof. The uniqueness is clear since (v, u1) = (v, u2) with v = u1 −u2 implies
∥u1 − u2∥
2 = (u1 − u2, u1 − u2) = 0. If L(v) = 0 for all v ∈ V , then we may
take u = 0. Assume now that L(v̄) ̸= 0 for some v̄ ∈ V . We will construct u
as a suitably normalized “normal vector” to the “hyperplane” V0 = {v ∈ V :
L(v) = 0}, which is easily seen to be a closed subspace of V , see Problem A.2.
Then v̄ = v0 + w with v0 ∈ V0 and w orthogonal to V0 and L(w) = L(v̄) ̸= 0.
But then L(v − w L(v)/L(w)) = 0, so that (v − w L(v)/L(w), w) = 0 and
hence L(v) = (v, u), ∀v ∈ V , where u = w L(w)/∥w∥2. ⊓.
This result makes it natural to identify the linear functionals L ∈ V ∗ with
the associated u ∈ V , and thus V ∗ is equivalent to V , in the case of a Hilbert
space.
We sometimes want to solve equations of the form: Find u ∈ V such that
(A.11) a(u, v) = L(v), ∀v ∈ V,
where V is a Hilbert space, L is a bounded linear functional on V , and a(·, ·)
is a symmetric bilinear form, which is coercive in V , i.e.,
(A.12) a(v, v) ≥ α∥v∥2V , ∀v ∈ V, with α > 0.
This implies that a(·, ·) is symmetric, positive definite, i.e., an inner product
on V , and the Riesz representation theorem immediately gives the existence
of a unique solution u ∈ V for each L ∈ V ∗.
Moreover, by taking v = u in (A.11) we get
α∥u∥2V ≤ a(u, u) = L(u) ≤ ∥L∥V ∗ ∥u∥V ,

A.1 Abstract Linear Spaces 229
so that, after cancelling one factor ∥u∥V ,
(A.13) ∥u∥V ≤ C∥L∥V ∗ , where C = 1/α.
This is an example of an energy estimate.
If a(·, ·) is a symmetric bilinear form, which is coercive and bounded in V ,
so that (A.12) and (A.9) hold, then we may define a norm ∥ · ∥a, the energy
norm, by
∥v∥a = a(v, v)
1/2, for v ∈ V,
By (A.12) and (A.9) we then have
(A.14)
√
α∥v∥V ≤ ∥v∥a ≤
√
M∥v∥V , ∀v ∈ V,
and thus the norm ∥ · ∥a on V is equivalent to ∥ · ∥V . Clearly, V is then also
a Hilbert space with respect to the scalar product a(·, ·) and norm ∥ · ∥a.
The solution of (A.11) may also be characterized in terms of a minimiza-
tion problem.
Theorem A.2. Assume that a(·, ·) is a symmetric, positive definite bilinear
form and that L is a bounded linear form on the Hilbert space V . Then u ∈ V
satisfies (A.11) if and only if
(A.15) F (u) ≤ F (v), ∀v ∈ V, where F (v) = 1
2
a(v, v) − L(v).
Proof. Suppose first that u satisfies (A.11). Let v ∈ V be arbitrary and define
w = v − u ∈ V . Then v = u + w and
F (v) = 1
2
a(u + w, u + w) − L(u + w)
= 1
2
a(u, u) − L(u) + a(u, w) − L(w) + 1
2
a(w, w)
= F (u) + 1
2
a(w, w),
where we have used (A.11) and the symmetry of a(·, ·). Since a is positive
definite, this proves (A.15).
Conversely, if (A.15) holds, then for v ∈ V given we have
g(t) := F (u + tv) ≥ F (u) = g(0), ∀ t ∈ R,
so that g(t) has a minimum at t = 0. But g(t) is the quadratic polynomial
g(t) = 1
2
a(u + tv, u + tv) − L(u + tv)
= 1
2
a(u, u) − L(u) + t
(
a(u, v) − L(v)
)
+ 1
2
t2a(v, v),
and thus 0 = g′(0) = a(u, v) − L(v), which is (A.11). ⊓.
Thus, u ∈ V satisfies (A.11) if and only if u minimizes the energy func-
tional F . This method of studying the minimization problem by varying the
argument of the functional F around the given vector u is called a variational
method, and the equation (A.11) is called the variational equation of F .
The following theorem, which is known as the Lax-Milgram lemma, ex-
tends the Riesz representation theorem to nonsymmetric bilinear forms.

230 A Some Tools from Mathematical Analysis
Theorem A.3. If the bilinear form a(·, ·) is bounded and coercive in the
Hilbert space V , and L is a bounded linear form in V , then there exists a
unique vector u ∈ V such that (A.11) is satisfied. Moreover, the energy esti-
mate (A.13) holds.
Proof. With (·, ·) the inner product in V we have by Riesz’ representation
theorem that there exists a unique b ∈ V such that
L(v) = (b, v), ∀v ∈ V.
Moreover, for each u ∈ V , a(u, ·) is clearly also a bounded linear functional
on V , so that there exists a unique A(u) ∈ V such that
a(u, v) = (A(u), v), ∀v ∈ V.
It is easy to check that A(u) depends linearly and boundedly on u, so that
Au = A(u) defines A : V → V as a bounded linear operator. The equation
(A.11) is therefore equivalent to Au = b, and to complete the proof of the
theorem we shall show that this equation has a unique solution u = A−1b for
each b.
Using the coercivity we have
α∥v∥2V ≤ a(v, v) = (Av, v) ≤ ∥Av∥V ∥v∥V ,
so that
(A.16) α∥v∥V ≤ ∥Av∥V , ∀v ∈ V.
This shows uniqueness, since Av = 0 implies v = 0. This may also be ex-
pressed by saying that the null space N (A) = {v ∈ V : Av = 0} = 0, or that
A is injective.
To show that there exists a solution u for each b ∈ V means to show that
each b ∈ V belongs to the range R(A) = {w ∈ V : w = Av for some v ∈ V },
i.e., R(A) = V , or A is surjective. To see this we first note that R(A) is a closed
linear subspace of V . To show that R(A) is closed, assume that Avj → w in
V as j → ∞. Then by (A.16) we have ∥vj − vi∥V ≤ α
−1∥Avj − Avi∥V → 0
as i, j → ∞. Hence vj → v ∈ V as j → ∞, and by the continuity of A, also
Avj → Av = w. Therefore, w ∈ R(A) and R(A) is closed.
Assume now that R(A) ̸= V . Then, by the projection theorem, there
exists w ̸= 0, which is orthogonal to R(A). But, by the orthogonality,
α∥w∥2V ≤ a(w, w) = (Aw, w) = 0,
so that w = 0, which is a contradiction. Hence R(A) = V . This completes
the proof that there is a unique solution for each b ∈ V . The energy estimate
is proved in the same way as before. ⊓.

A.2 Function Spaces 231
In the unsymmetric case there is no characterization of the solution in
terms of energy minimization.
We finally make a remark about linear equations in finite-dimensional
spaces. Let V = RN and consider a linear equation in V , which may be
written in matrix form as
Au = b,
where A is a N × N matrix and u, b are N -vectors. It is well-known that
this equation has a unique solution u = A−1b for each b ∈ V , if the matrix
A is nonsingular, i.e., if its determinant det(A) ̸= 0. If det(A) = 0, then the
homogeneous equation Au = 0 has nontrivial solutions u ̸= 0, and R(A) ̸= V
so that the inhomogeneous equation is not always solvable. Thus we have
neither uniqueness nor existence for all b ∈ V . In particular, uniqueness only
holds when det(A) ̸= 0, and we then also have existence. It is sometimes easy
to prove uniqueness, and we then also obtain the existence of the solution at
the same time.
A.2 Function Spaces
The Spaces Ck
For M ⊂ Rd we denote by C(M ) the linear space of continuous functions
on M . The subspace Cb(M ) of all bounded functions is made into a normed
linear space by setting (with a slight abuse of notation)
(A.17) ∥v∥C(M) = sup
x∈M
|v(x)|.
For example, this defines ∥v∥C(Rd), which we use frequently. When M is
a bounded and closed set, i.e., a compact set, the supremum in (A.17) is
attained in M and we may write
∥v∥C(M) = max
x∈M
|v(x)|.
The norm (A.17) is therefore called the maximum-norm. Note that conver-
gence in C(M ),
∥vi − v∥C(M) = sup
x∈M
|vi(x) − v(x)| → 0, as i → ∞,
is the same as uniform convergence in M . Recall that if a sequence of con-
tinuous functions is uniformly convergent in M , then the limit function is
continuous. Using this fact it is not difficult to prove that C(M ) is a complete
normed space, i.e., a Banach space. C(M ) is not a Hilbert space, because the
maximum-norm is not associated with a scalar product as in (A.1).

232 A Some Tools from Mathematical Analysis
Let now Ω ⊂ Rd be a domain, i.e., a connected open set. For any integer
k ≥ 0, we denote by Ck(Ω) the linear space of all functions v that are k times
continuously differentiable in Ω, and by Ck(Ω̄) the functions in Ck(Ω), for
which Dαv ∈ C(Ω̄) for all |α| ≤ k, where Dαv denotes the partial derivative
of v defined in (1.8). If Ω is bounded, then the latter space is a Banach space
with respect to the norm
∥v∥Ck(Ω̄) = max
|α|≤k
∥Dαv∥C(Ω̄).
For functions in Ck(Ω̄), k ≥ 1, we sometimes also use the seminorm containing
only the derivatives of highest order,
|v|Ck(Ω̄) = max
|α|=k
∥Dαv∥C(Ω̄).
A function has compact support in Ω if it vanishes outside some compact
subset of Ω. We write Ck0 (Ω) for the space of functions in C
k(Ω) with compact
support in Ω. In particular, such functions vanish near the boundary Γ , and
for very large x if Ω is unbounded.
We say that a function is smooth if, depending on the context, it has
sufficiently many continuous derivatives for the purpose at hand.
When there is no risk for confusion, we omit the domain of the functions
from the notation of the spaces and write, e.g., C for C(Ω̄) and ∥ · ∥Ck for
∥ · ∥Ck(Ω̄), and similarly for other spaces that we introduce below.
Integrability, the Spaces Lp
Let Ω be a domain in Rd. We shall need to work with integrals of functions
v = v(x) in Ω which are more general than those in C(Ω̄). For a nonnegative
function one may define the so-called Lebesgue integral
IΩ(v) =
∫
Ω
v(x) dx,
which may be either finite or infinite, and which agrees with the standard
Riemann integral for v ∈ C(Ω̄). The functions we consider are assumed mea-
surable; we shall not go into details about this concept but just note that
all functions that we encounter in this text will satisfy this requirement. A
nonnegative function v is said to be integrable if IΩ(v) < ∞, and a general real or complex-valued function v is similarly integrable if |v| is integrable. A subset Ω0 of Ω is said to be a nullset, or a set of measure 0, if its volume |Ω| equals 0. Two functions which are equal except on a nullset are said to be equal almost everywhere (a.e.), and they then have the same integral. Thus if v1(x) = 1 in a bounded domain Ω and if v2(x) = 1 in Ω except at x0 ∈ Ω where v2(x0) = 2, then IΩ(v1) = IΩ(v2) = |Ω|. In particular, from the fact that a function is integrable we cannot draw any conclusion about its value A.2 Function Spaces 233 at a point x0 ∈ Ω, i.e., the point values are not well defined. Also, since the boundary Γ of Ω is a nullset, IΩ̄(v) = IΩ(v) for any v. We now define ∥v∥Lp = ∥v∥Lp(Ω) = ⎧ ⎪⎨ ⎪⎩ (∫ Ω |v(x)|p dx )1/p , for 1 ≤ p < ∞, ess sup Ω |v(x)|, for p = ∞, and say that v ∈ Lp = Lp(Ω) if ∥v∥Lp < ∞. Here the ess sup means the essential supremum, disregarding values on nullsets, so that, e.g., ∥v2∥L∞ = 1 for the function v2 above, even though supΩ v2 = 2. One may show that Lp is a complete normed space, i.e., a Banach space; the triangle inequality in Lp is called Minkowski’s inequality. Clearly, any v ∈ C belongs to Lp for 1 ≤ p ≤ ∞ if Ω is bounded, and ∥v∥Lp ≤ C∥v∥C, with C = |Ω| 1/p, for 1 ≤ p < ∞, and ∥v∥L∞ = ∥v∥C, but Lp also contains functions that are not continuous. Moreover, it is not difficult to show that C(Ω̄) is incomplete with respect to the Lp-norm for 1 ≤ p < ∞. To see this one constructs a sequence {vi} ∞ i=1 ⊂ C(Ω̄), which is a Cauchy sequence with respect to the Lp-norm, i.e., such that ∥vi −vj∥Lp → 0, but whose limit v = limi→∞ vi is discontinuous. However, C(Ω̄) is a dense subspace of Lp(Ω) for 1 ≤ p < ∞, if Γ is sufficiently smooth. By this we mean that for any v ∈ Lp there is a sequence {vi} ∞ i=1 ⊂ C such that ∥vi − v∥Lp → 0 as i → ∞. In other words, any function v ∈ Lp can be ap- proximated arbitrarily well in the Lp-norm by functions in C (in fact, for any k by functions in Ck0 ). In contrast, C is not dense in L∞ since a discontinuous function cannot be well approximated uniformly by a continuous function. The case L2 is of particular interest to us, and this space is an inner product space, and hence a Hilbert space, with respect to the inner product (A.18) (v, w) = ∫ Ω v(x)w(x) dx. In the case of complex-valued functions one takes the complex conjugate of w(x) in the integrand. Sobolev Spaces We shall now introduce some particular Hilbert spaces which are natural to use in the study of partial differential equations. These spaces consist of functions which are square integrable together with their partial derivatives up to a certain order. To define them we first need to generalize the concept of a partial derivative. Let Ω be a domain in Rd and let first v ∈ C1(Ω̄). Integration by parts yields 234 A Some Tools from Mathematical Analysis ∫ Ω ∂v ∂xi φ dx = − ∫ Ω v ∂φ ∂xi dx, ∀φ ∈ C10 = C 1 0 (Ω). If v ∈ L2 = L2(Ω), then ∂v/∂xi does not necessarily exist in the classical sense, but we may define ∂v/∂xi to be the linear functional (A.19) L(φ) = ∂v ∂xi (φ) = − ∫ Ω v ∂φ ∂xi dx, ∀φ ∈ C10 . This functional is said to be a generalized or weak derivative of v. When L is bounded in L2, i.e., |L(φ)| ≤ C∥φ∥, it follows from Riesz’ representation theorem that there exists a unique function w ∈ L2, such that L(φ) = (w, φ) for all φ ∈ L2, and in particular − ∫ Ω v ∂φ ∂xi dx = ∫ Ω wφ dx, ∀φ ∈ C10 . We then say that the weak derivative belongs to L2 and write ∂v/∂xi = w. In this case we thus have (A.20) ∫ Ω ∂v ∂xi φ dx = − ∫ Ω v ∂φ ∂xi dx, ∀φ ∈ C10 . In particular, if v ∈ C1(Ω̄), then the generalized derivative ∂v/∂xi coincides with the classical derivative ∂v/∂xi. In a similar way, with Dαv denoting the partial derivative of v defined in (1.8), we define the weak partial derivative Dαv as the linear functional (A.21) Dαv(φ) = (−1)|α| ∫ Ω v Dαφ dx, ∀φ ∈ C |α| 0 . When this functional is bounded in L2, Riesz’ representation theorem shows that there exists a unique function in L2, which we denote by D αv, such that (Dαv, φ) = (−1)|α|(v, Dαφ), ∀φ ∈ C |α| 0 . We refer to Problem A.9 for further discussion of generalized functions. We now define Hk = Hk(Ω), for k ≥ 0, to be the space of all functions whose weak partial derivatives of order ≤ k belong to L2, i.e., Hk = Hk(Ω) = { v ∈ L2 : D αv ∈ L2 for |α| ≤ k } , and we equip this space with the inner product (v, w)k = (v, w)Hk = ∑ |α|≤k ∫ Ω DαvDαw dx, and the corresponding norm A.2 Function Spaces 235 ∥v∥k = ∥v∥Hk = (v, v) 1/2 Hk = ( ∑ |α|≤k ∫ Ω (Dαv)2 dx )1/2 . In particular, ∥v∥0 = ∥v∥L2 , and in this case we normally omit the subscript 0 and write ∥v∥. Also ∥v∥1 = (∫ Ω { v2 + d∑ j=1 ( ∂v ∂xj )2} dx )1/2 = ( ∥v∥2 + ∥∇v∥2 )1/2 and ∥v∥2 = (∫ Ω { v2 + d∑ j=1 ( ∂v ∂xj )2 + d∑ i,j=1 ( ∂2v ∂xi∂xj )2} dx )1/2 . We sometimes also use the seminorm, for k ≥ 1, (A.22) |v|k = |v|Hk = ( ∑ |α|=k ∫ Ω (Dαv)2 dx )1/2 . Note that the seminorm vanishes for constant functions. Using the fact that L2 is complete, one may show that H k is complete and thus a Hilbert space, see Problem A.4. The space Hk is an example of a more general class of function spaces called Sobolev spaces. It may be shown that Cl = Cl(Ω̄) is dense in Hk = Hk(Ω) for any l ≥ k, if Γ is sufficiently smooth. This is useful because it allows us to obtain certain results for Hk by carrying out the proof for functions in Ck, which may be technically easier, and then extend the result to all v ∈ Hk by using the density, cf. the proof of Theorem A.4 below. Similarly, we denote by W kp = W k p (Ω) the normed space defined by the norm ∥v∥W kp = (∫ Ω ∑ |α|≤k |Dαv|p dx )1/p , for 1 ≤ p < ∞, with the obvious modification for p = ∞. This space is in fact complete and hence a Banach space. For p = 2 we have W k2 = H k. Again, for v ∈ Ck we have ∥v∥W k∞ = ∥v∥Ck . Trace Theorems If v ∈ C(Ω̄), then v(x) is well defined for x ∈ Γ , the boundary of Ω. The trace γv of such a v on Γ is the restriction of v to Γ , i.e., (A.23) (γv)(x) = v(x), for x ∈ Γ. Recall that since Γ is a nullset, the trace of v ∈ L2(Ω) is not well defined. Suppose now that v ∈ H1(Ω). Is it then possible to define v uniquely on Γ , i.e., to define its trace γv on Γ ? (One may show that functions in 236 A Some Tools from Mathematical Analysis H1(Ω) are not necessarily continuous, cf. Theorem A.5 and Problems A.6, A.7 below.) This question can be made more precise by asking if it possible to find a norm ∥ · ∥(Γ ) for functions on Γ and some constant C that (A.24) ∥γv∥(Γ ) ≤ C∥v∥1, ∀v ∈ C 1(Ω̄). An inequality of this form is called a trace inequality. If (A.24) holds, then by a density argument (see below) it is possible to extend the domain of definition of the trace operator γ from C1(Ω̄) to H1(Ω), and (A.24) will also hold for all v ∈ H1(Ω). The function space to which γv will belong will be defined by the norm ∥ · ∥(Γ ) in (A.24). We remark that in the above discussion the boundary Γ could be replaced by some other subset of Ω of dimension smaller than d. In order to proceed with the trace theorems, we first consider a one- dimensional case, with Γ corresponding to a single point. Lemma A.1. Let Ω = (0, 1). Then there is a constant C such that |v(x)| ≤ C∥v∥1, ∀x ∈ Ω̄, ∀v ∈ C 1(Ω̄). Proof. For x, y ∈ Ω we have v(x) = v(y) + ∫ x y v′(s) ds, and hence by the Cauchy-Schwarz inequality |v(x)| ≤ |v(y)| + ∫ 1 0 |v′(s)| ds ≤ |v(y)| + ∥v′∥. Squaring both sides and integrating with respect to y, we obtain, (A.25) v(x)2 ≤ 2 ( ∥v∥2 + ∥v′∥2 ) = 2∥v∥21. which shows the desired estimate. ⊓. We now show a simple trace theorem. By L2(Γ ) we denote the Hilbert space of all functions that are square integrable on Γ with norm ∥w∥L2(Γ ) = (∫ Γ w2 ds )1/2 . Theorem A.4. (Trace theorem.) Let Ω be a bounded domain in Rd (d ≥ 2) with smooth or polygonal boundary Γ . Then the trace operator γ : C1(Ω̄) → C(Γ ) may be extended to γ : H1(Ω) → L2(Γ ), which defines the trace γv ∈ L2(Γ ) for v ∈ H 1(Ω). Moreover, there is a constant C = C(Ω) such that (A.26) ∥γv∥L2(Γ ) ≤ C∥v∥1, ∀v ∈ H 1(Ω). Proof. We first show the trace inequality (A.26) for functions v ∈ C1(Ω̄). For simplicity we consider only the case when Ω = (0, 1) × (0, 1), the unit square A.2 Function Spaces 237 in R2. The proof in the general case is similar. For x = (x1, x2) ∈ Ω we have by (A.25) v(0, x2) 2 ≤ 2 (∫ 1 0 v(x1, x2) 2 dx1 + ∫ 1 0 ( ∂v ∂x1 (x1, x2) )2 dx1 ) , and hence by integration with respect to x2, ∫ 1 0 v(0, x2) 2 dx2 ≤ 2 ( ∥v∥2 + ∥∇v∥2 ) = 2∥v∥21. The analogous estimates for the remaining parts of Γ complete the proof of (A.26) for v ∈ C1. Let now v ∈ H1(Ω). Since C1 is dense in H1 there is a sequence {vi} ∞ i=1 ⊂ C1 such that ∥v − vi∥1 → 0. This sequence is then a Cauchy sequence in H 1, i.e., ∥vi − vj∥1 → 0 as i, j → ∞. Applying (A.26) to vi − vj ∈ C 1, we find ∥γvi − γvj∥L2(Γ ) ≤ C∥vi − vj∥1 → 0, as i, j → ∞, i.e., {γvi} ∞ i=1 is a Cauchy sequence in L2(Γ ), and thus there exists w ∈ L2(Γ ) such that γvi → w in L2(Γ ) as i → ∞. We define γv = w. It is easy to show that (A.26) then holds for v ∈ H1. This extends γ to a bounded linear operator γ : H1(Ω) → L2(Γ ). Since C 1 is dense in H1, there is only one such extension (prove this!). In particular, γ is independent of the choice of the sequence {vi}. ⊓. The constant in Theorem A.4 depends on the size and shape of the domain Ω. It is sometimes important to have more detailed information about this dependence, and in Problem A.15 we assume that the shape is fixed (a square) and investigate the dependence of the constant on the size of Ω. The following result, of a somewhat similar nature, is a special case of the well-known and important Sobolev inequality. Theorem A.5. Let Ω be a bounded domain in Rd with smooth or polygonal boundary and let k > d/2. Then Hk(Ω) ⊂ C(Ω̄), and there exists a constant
C = C(Ω) such that
(A.27) ∥v∥C ≤ C∥v∥k, ∀v ∈ H
k(Ω).
In the same way as for the trace theorem it suffices to show (A.27) for
smooth v, see Problem A.20. The particular case when d = k = 1 is given
in Lemma A.1, and Problem A.13 considers the case Ω = (0, 1) × (0, 1). The
general case is more complicated. As shown in Problems A.6, A.7, a function
in H1(Ω) with Ω ⊂ Rd is not necessarily continuous when d ≥ 2.
If we apply Sobolev’s inequality to derivatives of v, we get
(A.28) ∥v∥Cℓ ≤ C∥v∥k, ∀v ∈ H
k(Ω), if k > ℓ + d/2,
and we may similarly conclude that Hk(Ω) ⊂ Cℓ(Ω̄) if k > ℓ + d/2.

238 A Some Tools from Mathematical Analysis
The Space H1
0
(Ω). Poincaré’s Inequality
Theorem A.4 shows that the trace operator γ : H1(Ω) → L2(Γ ) is a bounded
linear operator. This implies that its null space,
H10 (Ω) =
{
v ∈ H1(Ω) : γv = 0
}
,
is a closed subspace of H1(Ω), and hence a Hilbert space with the norm ∥·∥1.
It is the set of functions in H1 that vanish on Γ in the sense of trace. We
note that the seminorm |v|1 = ∥∇v∥ defined in (A.22) is in fact a norm on
H10 (Ω), equivalent to ∥ · ∥1, as follows from the following result.
Theorem A.6. (Poincaré’s inequality.) If Ω is a bounded domain in Rd,
then there exists a constant C = C(Ω) such that
(A.29) ∥v∥ ≤ C∥∇v∥, ∀v ∈ H10 (Ω).
Proof. As an example we show the result for Ω = (0, 1) × (0, 1). The proof
in the general case is similar.
Since C10 is dense in H
1
0 , it suffices to show (A.29) for v ∈ C
1
0 . For such a
v we write
v(x) =
∫ x1
0
∂v
∂x1
(s, x2) ds, for x = (x1, x2) ∈ Ω,
and hence by the Cauchy-Schwarz inequality
|v(x)|2 ≤
∫ 1
0
ds
∫ 1
0
( ∂v
∂x1
(s, x2)
)2
ds.
The result now follows by integration with respect to x2 and x1, with C = 1
in this case. ⊓.
The equivalence of the norms | · |1 and ∥ · ∥1 on H
1
0 (Ω) now follows from
∥∇v∥2 ≤ ∥v∥21 = ∥v∥
2 + ∥∇v∥2 ≤ (C + 1)∥∇v∥2, ∀v ∈ H10 (Ω).
The dual space of H10 (Ω) is denoted H
−1(Ω). Thus H−1 = (H10 )
∗ is the
space of all bounded linear functionals on H10 . The norm in H
−1 is (cf. (A.8))
(A.30) ∥L∥(H1
0
)∗ = ∥L∥−1 = sup
v∈H1
0
|L(v)|
|v|1
.
A.3 The Fourier Transform
Let v be a real or complex function in L1(R
d). We define its Fourier transform
for ξ = (ξ1, . . . , ξd) ∈ R
d by

A.3 The Fourier Transform 239
Fv(ξ) = v̂(ξ) =
∫
Rd
v(x)e−ix·ξ dx, where x · ξ =
d∑
j=1
xj ξj .
The inverse Fourier transform is
F
−1v(x) = v̌(x) = (2π)−d
∫
Rd
v(ξ)eix·ξ dξ = (2π)−dv̂(−x), for x ∈ Rd.
If v and v̂ are both in L1(R
d), then Fourier’s inversion formula holds, i.e.,
F
−1(F v) = (v̂)̌ = v.
The inner product in L2(R
d) of two functions can be expressed in terms of
their Fourier transforms according to Parseval’s formula,
(A.31)
∫
Rd
v(x)w(x) dx = (2π)−d
∫
Rd
v̂(ξ)ŵ(ξ) dξ,
or
(v, w) = (2π)−d(v̂, ŵ), where (v, w) = (v, w)L2(Rd).
In particular, we have for the corresponding norms
(A.32) ∥v∥ = (2π)−d/2∥v̂∥.
Let Dαv be a partial derivative of v as defined in (1.8). We then have,
assuming that v and its derivatives are sufficiently small for |x| large,
F(Dαv)(ξ) = (iξ)αv̂(ξ) = i|α|ξαv̂(ξ), where ξα = ξα11 · · · ξ
αd
d .
In fact, by integration by parts,
∫
Rd
Dαv(x)e−ix·ξ dx = (−1)|α|
∫
Rd
v(x)Dα(e−ix·ξ) dx = (iξ)αv̂(ξ).
Further, translation of the argument of the function corresponds to multipli-
cation of its Fourier transform by an exponential,
(A.33) Fv(· + y)(ξ) = eiy·ξv̂(ξ), for y ∈ Rd,
and for scaling of the argument we have
(A.34) Fv(a·)(ξ) = a−dv̂(a−1ξ), for a > 0.
The convolution of two functions v and w is defined by
(v ∗ w)(x) =
∫
Rd
v(x − y)w(y) dy =
∫
Rd
v(y)w(x − y) dy,
and we have

240 A Some Tools from Mathematical Analysis
F(v ∗ w)(ξ) = v̂(ξ)ŵ(ξ),
because
∫
Rd
(∫
Rd
v(x − y)w(y) dy
)
e−ix·ξ dx
=
∫
Rd
∫
Rd
v(x − y)w(y)e−i(x−y)·ξe−iy·ξ dx dy
=
∫
Rd
∫
Rd
v(z)w(y)e−iz·ξe−iy·ξ dz dy.
It follows, which can also easily be shown directly, that differentiation of a
convolution can be carried out on either factor,
Dα(v ∗ w) = Dαv ∗ w = v ∗ Dαw.
A.4 Problems
Problem A.1. Let V be a Hilbert space with scalar product (·, ·) and let
u ∈ V be given. Define L : V → R by L(v) = (u, v) ∀v ∈ V . Prove that L is
a bounded linear functional on V . Determine ∥L∥.
Problem A.2. Prove that if L : V → R is a bounded linear functional and
{vi} is a sequence with L(vi) = 0 that converges to v ∈ V , then L(v) = 0.
This proves that the subspace V0 in the proof of Theorem A.1 is closed.
Problem A.3. Prove the energy estimate (A.13) by using (A.10) and (A.14).
Hint: Recall (A.8) and note that (A.10) means
sup
v∈V \{0}
|L(v)|
∥v∥a
= ∥u∥a.
Problem A.4. Given that L2(Ω) is complete, prove that H
1(Ω) is complete.
Hint: Assume that ∥vj − vi∥1 → 0 as i, j → ∞. Show that there are v, wk
such that ∥vj − v∥ → 0, ∥∂vj /∂xk − wk∥ → 0, and that wk = ∂v/∂xk in the
sense of weak derivative.
Problem A.5. Let Ω = (−1, 1) and let v : Ω → R be defined by v(x) = 1 if
x ∈ (−1, 0) and v(x) = 0 if x ∈ (0, 1). Prove that v ∈ L2(Ω) and that v can
be approximated arbitrarily well in L2-norm by C
0-functions.
Problem A.6. Let Ω be the unit ball in Rd, d = 1, 2, 3, i.e., Ω = {x ∈ Rd :
|x| < 1}. For which values of λ ∈ R does the function v(x) = |x|λ belong to (a) L2(Ω), (b) H 1(Ω)? Problem A.7. Check if the function v(x) = log(− log |x|2) belongs to H1(Ω) if Ω = {x ∈ R2 : |x| < 1 2 }. Are functions in H1(Ω) necessarily bounded and continuous? A.4 Problems 241 Problem A.8. It is known that C10 (Ω) is dense in L2(Ω) and H 1 0 (Ω). Explain why C10 (Ω) is not dense in H 1(Ω). Problem A.9. The generalized (or weak) derivative defined in (A.19) is a special case of the so-called generalized functions or distributions. Another important example is Dirac’s delta, which is defined as a linear functional acting on continuous test functions, for Ω ⊂ Rd, δ(φ) = φ(0), ∀φ ∈ C00 (Ω). Let now d = 1, Ω = (−1, 1) and f (x) = { x, x ≥ 0, 0, x ≤ 0, g(x) = { 1, x > 0,
0, x < 0. Show that f ′ = g, g′ = δ in the sense of generalized derivative, i.e., f ′(φ) = − ∫ Ω f φ′ dx = ∫ Ω gφ dx, ∀φ ∈ C10 (Ω), g′(φ) = − ∫ Ω gφ′ dx = φ(0), ∀φ ∈ C10 (Ω). Conclude that the generalized derivative f ′ = g belongs to L2, but that g ′ = δ does not. For the latter statement, you must show that δ is not bounded with respect to the L2-norm, i.e., you need to find a sequence of test functions such that ∥φi∥L2 → 0, but φi(0) = 1 as i → ∞. Thus, f ∈ H 1(Ω) and g ̸∈ H1(Ω). Problem A.10. For f ∈ L2(Ω) we define the linear functional f (v) = (f, v) ∀v ∈ L2(Ω). Show the inequality, cf. (A.30), ∥f∥−1 ≤ C∥f∥, ∀f ∈ L2(Ω). Conclude that L2(Ω) ⊂ H −1(Ω). Problem A.11. Let Ω = (0, 1) and f (x) = 1/x. Show that f ̸∈ L2(Ω). Show that f ∈ H−1(Ω) by defining the linear functional f (v) = (f, v) ∀v ∈ H10 (Ω), and proving the inequality |(f, v)| ≤ C∥v′∥, ∀v ∈ H10 (Ω). Conclude that H−1(Ω) ̸⊂ L2(Ω). Problem A.12. Prove that if Ω = (0, L) is a finite interval, then there is a constant C = C(L) such that, for all x ∈ Ω̄ and v ∈ C1(Ω̄), |v(x)| ≤ L−1 ∫ Ω |v| dy + ∫ Ω |v′| dy ≤ C∥v∥W 1 1 (Ω),(a) |v(x)|2 ≤ L−1 ∫ Ω |v|2 dy + L ∫ Ω |v′|2 dy ≤ C∥v∥21,(b) |v(x)|2 ≤ L−1∥v∥2 + 2∥v∥ ∥v′∥ ≤ C∥v∥ ∥v∥1.(c) 242 A Some Tools from Mathematical Analysis Problem A.13. Prove that if Ω is the unit square in R2, then there exists a constant C such that ∥v∥L1(Γ ) ≤ C∥v∥W 11 (Ω), ∀v ∈ C 1(Ω̄),(a) ∥v∥C ≤ C∥v∥W 2 1 , ∀v ∈ C2(Ω̄).(b) Since ∥v∥W 2 1 ≤ 31/2|Ω|1/2∥v∥H2 , part (b) implies the special case of Theorem A.5 with k = d = 2 and Ω a square domain. Part (b) directly generalizes to ∥v∥C ≤ C∥v∥W d 1 for Ω ⊂ Rd. Hint: Proof of Theorem A.4. Problem A.14. (Scaling of Sobolev norms.) Let L be a positive number and consider the coordinate transformation x = Lx̂, which maps the bounded domain Ω ⊂ Rd onto Ω̂. A function v defined on Ω is transformed to a function v̂ on Ω̂ according to v̂(x̂) = v(Lx̂). Prove the scaling identities ∥v∥L2(Ω) = L d/2 ∥v̂∥L2(Ω̂),(a) ∥∇v∥L2(Ω) = L d/2−1 ∥∇̂v̂∥L2(Ω̂),(b) ∥v∥L2(Γ ) = L d/2−1/2 ∥v̂∥L2(Γ̂ ).(c) Problem A.15. (Scaled trace inequality.) Let Ω = (0, L)×(0, L) be a square domain of side L. Prove the scaled trace inequality ∥v∥L2(Γ ) ≤ C ( L−1∥v∥2L2(Ω) + L∥∇v∥ 2 L2(Ω) )1/2 , ∀v ∈ C1(Ω̄). Hint: Apply (A.26) with Ω̂ = (0, 1) × (0, 1) and use the scaling identities in Problem A.14. Problem A.16. Let Ω be the unit square in R2. Prove the trace inequality in the form ∥v∥2L2(Γ ) ≤ C ( ∥v∥2L2(Ω) + ∥v∥L2(Ω)∥∇v∥L2(Ω) ) ≤ C∥v∥ ∥v∥1. Hint: Start from v(0, y2) 2 = v(y1, y2) 2 − ∫ y1 0 ∂ ∂x1 v(s, y2) 2 ds. Problem A.17. It is a fact from linear algebra that all norms on a finite- dimensional space V are equivalent. Illustrate this by proving the following norm equivalences in V = RN : ∥v∥l2 ≤ ∥v∥l1 ≤ √ N∥v∥l2 ,(A.35) ∥v∥l∞ ≤ ∥v∥l2 ≤ √ N∥v∥l∞ ,(A.36) ∥v∥l∞ ≤ ∥v∥l1 ≤ N∥v∥l∞ ,(A.37) A.4 Problems 243 where ∥v∥lp = ( N∑ j=1 |vj| p )1/p for 1 ≤ p < ∞, ∥v∥l∞ = max 1≤j≤N |vj|. Note that the equivalence constants tend to infinity as N → ∞. Problem A.18. Prove (A.33) and (A.34). Problem A.19. Prove that the Fourier transform of v(x) = e−|x| 2 is v̂(ξ) = πd/2e−|ξ| 2/4. Problem A.20. Assume that Sobolev’s inequality in (A.27) has been proved for all v ∈ Ck(Ω̄) with k > d/2. Prove Sobolev’s imbedding Hk(Ω) ⊂ C(Ω̄).
In other words, for each v ∈ Hk(Ω) show that there is w ∈ C(Ω̄) such that
v = w almost everywhere, i.e., ∥v − w∥L2 = 0. Hint: C
k(Ω̄) is dense in Hk(Ω)
and C(Ω̄) is a Banach space.

B Orientation on Numerical Linear Algebra
Both finite difference and finite element methods for elliptic problems lead
to linear algebraic systems of equations of the form
(B.1) AU = b,
where A is a nonsingular square matrix of order N . Also in time-stepping
methods for evolution equations, problems of elliptic type need to be solved
in the successive time steps. To solve such systems efficiently therefore be-
comes an important part of numerical analysis. When the dimension of the
computational domain is at least 2 this is normally not possible by direct
methods, and, except in special cases, one therefore turns to iterative meth-
ods. These take advantage of the fact that the matrices involved are sparse,
i.e., most of their elements are zero, and have other special features. In this
appendix we give a short overview, without proofs, of the most commonly
used methods.
B.1 Direct Methods
We consider first the case that the system (B.1) derives from the standard
finite difference approximation (4.3) of the two-point boundary value problem
(4.1). In this case A is a tridiagonal matrix, and it is easy to see that A
may then be factored in O(N ) algebraic operations as A = LR, where L is
bidiagonal and lower triangular and R is bidiagonal, upper triangular. The
system may thus be written
LRU = b,
and one may now first solve LG = b for G = RU in O(N ) operations and
then solve the latter equation for U , also in O(N ) operations. Altogether
this is a direct method for (B.1), which requires O(N ) operations. Since the
number of unknowns is N , this is the smallest possible order for any method.
Consider now an elliptic problem in a domain Ω ⊂ Rd with d ≥ 2.
Using either finite differences or finite elements based on a quasi-uniform
family of meshes, the dimension N of the corresponding finite dimensional
problem is of order O(h−d), where h is the mesh-size, and for d ≥ 2 direct
solution by Gauss elimination is normally not feasible as this method requires

246 B Orientation on Numerical Linear Algebra
O(N 3) = O(h−3d) algebraic operations. Except in special cases one therefore
turns to iterative methods.
One case when a direct method can be used, however, is provided by the
model problem (4.11) with the five-point finite difference scheme on the unit
square, which may be solved directly by using the discrete Fourier transform,
defined by
b̂m =
∑
j
bj e
−2πi m·j h, m = (m1, m2), j = (j1, j2).
In fact, we then have (−∆hU )̂m = 2π
2|m|2Ûm, hence Ûm = (2π
2|m|2)−1b̂m,
so that by the inverse discrete Fourier transform
U j =
∑
m
(2π2|m|2)−1b̂me
2πi m·j h.
Using the Fast Fourier Transform (FFT) both b̂m and U
j may be calculated
in O(N log N ) operations.
B.2 Iterative Methods. Relaxation, Overrelaxation,
and Acceleration
As a basic iterative method for (B.1) we consider the Richardson method
(B.2) U n+1 = U n − τ (AU n − b) for n ≥ 0, with U 0 given,
where τ is a positive parameter. With U the exact solution of (B.1) we have
U n − U = R(U n−1 − U ) = · · · = Rn(U 0 − U ), where R = I − τ A,
and hence the rate of convergence of the method depends on ∥Rn∥, where
∥M∥ = max∥x∥=1 ∥M x∥ is the matrix norm subordinate to the Euclidean
norm ∥ · ∥ in RN . When A is symmetric positive definite (SPD) and has
eigenvalues {λj}
N
j=1, then, since {1 − τ λj}
N
j=1 are the eigenvalues of R, we
have
∥Rn∥ = ρn, where ρ = ρ(R) = max
i
|1 − τ λi|,
and (B.2) converges if ρ < 1. The choice of τ which gives the smallest value of ρ is τ = 2/(λ1 + λN ), in which case ρ = (κ − 1)/(κ + 1), where κ = κ(A) = λN /λ1 is the condition number or A. We note, however, that this choice of τ requires knowledge of λ1 and λN which is not normally available. In applications to second order elliptic problems one often has κ = O(h−2) so that ρ ≤ 1 − ch2 with c > 0. Hence with the optimal choice of τ the
number of iterations required to reduce the error to a small ϵ > 0 is of order
O(h−2| log ϵ|). Since each iteration uses O(h−d) operations in the application

B.2 Iterative Methods. Relaxation, Overrelaxation, and Acceleration 247
of I − τ A, this shows that the total number of operations needed to reduce
the error to a given tolerance is of order O(h−d−2), which is smaller than for
the direct solution by Gauss elimination when d ≥ 2.
The early more refined methods were designed for finite difference meth-
ods of positive type for second order elliptic equations, particularly for the
five-point scheme (4.12). The corresponding matrix may then be written
A = D − E − F , where D is diagonal and E and F are (elementwise) non-
negative and strictly lower and upper triangular. Examples of more efficient
methods are then the Jacobi and Gauss-Seidel methods which are defined by
(B.3) U n+1 = U n − B(AU n − b) = RU n + Bb, with R = I − BA,
in which B = BJ = D
−1 or B = BGS = (D − E)
−1, so that R =
RJ = D
−1(E + F ) and R = RGS = (D − E)
−1F , respectively. In the
application to the model problem (4.9) in the unit square, using the five-
point operator, the equations may be normalized so that D = 4I and
the application of RJ then simply means that the new value in the itera-
tion step at any interior mesh-point xj is obtained by taking the average
of the old values at the four neighboring points xj±el . The Gauss-Seidel
method also takes averages, but with the mesh-points taken in a given or-
der, and successively uses the values already obtained in forming the aver-
ages. The methods are therefore also referred to as the methods of simul-
taneous and successive displacements, respectively. For the model problem
one may easily determine the eigenvalues and eigenvectors of A and show
that with h = 1/M one has ρ(RJ) = cos(πh) = 1 −
1
2
π2h2 + O(h4) and
ρ(RGS) = ρ(RJ)
2 = 1−π2h2 +O(h4), so that the number of iterations needed
to reduce the error to ϵ is of the orders 2h−2π2| log ϵ| and h−2π2| log ϵ|, respec-
tively. The Gauss-Seidel method thus requires about half as many iterations
as the Jacobi method.
Forming the averages in the Jacobi and Gauss-Seidel methods may be
thought as relaxation. It turns out that one may obtain better results than
those described above by overrelaxation, i.e., by choosing
Bω = (D − ωE)
−1 and Rω = (D − ωE)
−1((1 − ω)E + F ), with ω > 1.
It may be shown that for the model problem the optimal choice of the para-
meter is
ωopt = 2/(1 +
√
1 − ρ2), where ρ = ρ(BJ) = cos(πh),
i.e., ωopt = 2/(1 + sin(πh)) = 2 − 2πh + O(h
2), and that correspondingly
ρ(Rωopt ) = ωopt − 1 = 1 − 2πh + O(h
2).
The number of iterations required is thus then of order O(h−1), which is
significantly smaller than for the above methods. This is the method of suc-
cessive overrelaxation (SOR).

248 B Orientation on Numerical Linear Algebra
We consider again an iterative method of the form (B.3) with ρ(R) < 1. For the purpose of accelerating the convergence we now introduce the new sequence V n = ∑n j=0 βnj U j , where the βnj are real numbers. Setting pn(λ) = ∑n j=0 βnj λ j , and assuming pn(1) = ∑n j=0 βnj = 1 for n ≥ 0, we obtain easily V n − U = pn(R)(U 0 − U ), where U is the solution of (B.1). For V n to converge fast to U one therefore wants to choose the βnj in such a way that the spectral radius ρ(pn(R)) becomes small with n. By the Cayley- Hamilton theorem for matrices one has pN (R) = 0, if pN is the characteristic polynomial of R, and hence V N = U , but this requires a prohibitively large number of iterations. For n < N we have by the spectral mapping theorem that ρ(pn(R)) = maxi |pn(µi)|, where {µi} N i=1 are the eigenvalues of R. In particular, if R is symmetric and ρ = ρ(R), so that |µi| ≤ ρ for all i, then one may show that, taking the maximum instead over [−ρ, ρ] ⊃ σ(R), the optimal polynomial is pn(λ) = Tn(λ/ρ)/Tn(1/ρ), where Tn is the nth Chebyshev polynomial, and the corresponding value of ρ(pn(R)) is bounded by Tn(1/ρ) −1 = 2 {(1 + √ 1 − ρ2 ρ )n + (1 + √ 1 − ρ2 ρ )−n}−1 ≤ 2 ( ρ 1 + √ 1 − ρ2 )n . For the model problem using the Gauss-Seidel basic iteration we have as above ρ = 1−π2h2+O(h4) and we find that the average error reduction factor per iteration step in our present method is bounded by 1 − √ 2πh + O(h2), which is of the same order of magnitude as for SOR. B.3 Alternating Direction Methods We now describe the Peaceman-Rachford alternating direction implicit it- erative method for the model problem (4.9) on the unit square, using the five-point discrete elliptic equation (4.11) with h = 1/M . In this case we may write A = H + V , where H and V correspond to the horizontal and vertical difference operators −h2∂1∂̄1 and −h 2∂2∂̄2. Note that H and V are positive definite and commute. Introducing an acceleration parameter τ and an inter- mediate value U n+1/2, we may consider the scheme defining U n+1 from U n by (B.4) (τ I + H)U n+1/2 = (τ I − V )U n + b, (τ I + V )U n+1 = (τ I − H)U n+1/2 + b, or after elimination, with Gτ appropriate and using that H and V commute, U n+1 = Rτ U n + Gτ , where Rτ = (τ I − H)(τ I + H) −1(τ I − V )(τ I + V )−1. B.4 Preconditioned Conjugate Gradient Methods 249 Note that the equations in (B.4) have tridiagonal matrices and may be solved in O(N ) operations, as we have indicated earlier. The error satisfies U n −U = Rnτ (U 0 − U ), and with µi the (common) eigenvalues of H and V , we have ∥Rτ ∥ ≤ maxi |(τ −µi)/(τ +µi)| 2 < 1, where it is easy to see that the maximum occurs for i = 1 or M . With µ1 = 4 sin 2( 1 2 πh), µM = 4 cos 2( 1 2 πh) the optimal τ is τopt = (µ1µM ) 1/2 with the maximum for i = 1, so that, with κ = κ(H) = κ(V ) = µM /µ1, ∥Rτopt∥ ≤ ((µ1µM )1/2 − µ1 (µ1µM )1/2 + µ1 )1/2 = κ1/2 − 1 κ1/2 + 1 = 1 − πh + O(h2). This again shows the same order of convergence as for SOR. A more efficient procedure is obtained by using varying acceleration pa- rameters τj , j = 1, 2, . . . , corresponding to the n step error reduction matrix R̃n = ∏n j=1 Rτj . It can be shown that the τj can be chosen cyclically with period m in such a way that m ≈ c log κ ≈ c log(1/h), so that the average error reduction rate is ∥R̃m∥ 1/m = max 1≤i≤M ( m−1∏ j=0 ∣ ∣τj − µi τj + µi ∣ ∣ )2/m ≤ 1 − c(log(1/h))−1, c > 0.
The analysis indicated depends strongly on the fact that H and V commute,
which only happens for rectangles and constant coefficients, but the method
may be defined and shown convergent in more general cases.
B.4 Preconditioned Conjugate Gradient Methods
We now turn to some iterative methods for systems mainly associated with
the emergence of the finite element method. We begin by describing the con-
jugate gradient method, and assume that A is SPD. Considering the iterative
method for (B.1) defined by
U n+1 = (I − τnA)U
n + τnb for n ≥ 0, with U
0 = 0,
we find at once that, for any choice of the parameters τj , U
n belongs to the
so-called Krylov space Kn(A; b) = span{b, Ab, . . . , A
n−1b}, i.e., consisting of
linear combinations of the Aib, i = 0, . . . , n − 1. The conjugate gradient
method defines these parameters so that U n is the best approximation of the
exact solution U of (B.1) in Kn(A; b) with respect to the norm defined by
|U| = (AU, U )1/2, i.e., U n is the orthogonal projection of U onto Kn(A; b)
with respect to the inner product (AV, W ). By our above discussion it follows
that, with κ = κ(A) the condition number of A,
(B.5) |U n − U| ≤ (Tn(1/ρ))
−1
|U| ≤ 2
(κ1/2 − 1
κ1/2 + 1
)n
|U|.

250 B Orientation on Numerical Linear Algebra
The computation of U n can be carried out by a two term recurrence
relation, for instance, in the following form using the residuals rn = b − AU n
and the auxiliary vectors qn ∈ Kn+1(A; b), orthogonal to Kn(A; b),
U n+1 = U n+
(rn, qn)
(Aqn, qn)
qn, qn+1 = rn+1−
(Arn+1, qn)
(Aqn, qn)
qn, U 0 = 0, q0 = b.
In the preconditioned conjugate gradient (PCG) method the conjugate
gradient method is applied to equation (B.1) after multiplication by some
SPD approximation B of A−1, which is easier to determine than A−1, so
that the equation (B.1) may be written BAU = Bb. We note that BA is
SPD with respect to the inner product (B−1V, W ). The error estimate (B.5)
is now valid in the corresponding norm with κ = κ(BA); B would be chosen
so that this condition number is smaller than κ(A). For the recursion formulas
the only difference is that now rn = B(b − AU n) and q0 = Bb.
B.5 Multigrid and Domain Decomposition Methods
In the case that the system (B.1) comes from a standard finite element prob-
lem, one way of defining a preconditioner as an approximate inverse of A is by
means of the multigrid method. This method is based on the observation that
large components of the errors are associated with low frequencies in a spec-
tral representation. The basic idea is then to work in a systematic way with
a sequence of triangulations and to reduce the low frequency errors on coarse
triangulations, which corresponds to small size problems, and to reduce the
higher frequency, or oscillatory, residual errors on finer triangulations by a
smoothing operator, such as a step of the Jacobi method, which is relatively
inexpensive.
Assuming that Ω is a plane polygonal domain we may, for instance, pro-
ceed as follows. We first perform a coarse triangulation of Ω. Each of the
triangles is then divided into four similar triangles, and this process is re-
peated, which after a finite number M of steps leads to a fine triangulation
with each of the original triangles divided into 4M small triangles. It is on
this fine triangulation which we want to use the finite element method, and
thus to define an iterative method. To find the next iterate U n+1 from U n we
start at the finest triangulation and go recursively from one level of fineness
to the previous in three steps:
1. A preliminary smoothing on the finer of the present triangulations.
2. Correction on the coarser triangulation by solving a residual equation.
3. A postsmoothing on the finer triangulation.
The execution of step 2 is thus itself carried out in three steps, starting with
a smoothing on the present level and going to step 2 on the next coarser level,
until one arrives at the original coarse triangulation, where the corresponding

B.5 Multigrid and Domain Decomposition Methods 251
residual equation is solved exactly. Postsmoothing on successive finer levels
then completes the algorithm for computing the next iterate U n+1. This
particular procedure is referred to as the V-cycle algorithm. It turns out that,
under the appropriate assumptions, the error reduction matrix R satisfies
∥R∥ ≤ ρ < 1, with ρ independent of M , i.e., of h, and that the number of operations is of order O(N ), where N = O(h−2) is the dimension of the matrix associated with the finest triangulation. A class of iterative methods that have attracted a lot of attention recently is the so called domain decomposition methods. These assume that the do- main Ω in which we want to solve our elliptic problem may be decomposed into subdomains Ωj , j = 1, . . . , M, which could overlap. The idea is to reduce the boundary value problem on Ω into problems on each of the Ωj , which are then coupled by their values on the intersections. The problems on the Ωj could be solved independently on parallel processors. This is particularly efficient when the individual problems may be solved very fast, e.g., by fast transform methods. The domain decomposition methods go back to the Schwarz alternating procedure, in which Ω = Ω1 ∪ Ω2 for two overlapping domains Ω1 and Ω2. Considering the Dirichlet problem (1.1) and (1.2) on Ω (with g = 0 on Γ ) one defines a sequence {uk}∞k=0 starting with a given u 0 vanishing on ∂Ω, by −∆u2k+1 = f in Ω1, u2k+1 = { u2k on ∂Ω1 ∩ Ω2, 0 on ∂Ω1 ∩ ∂Ω, −∆u2k+2 = f in Ω2, u2k+2 = { u2k+1 on ∂Ω2 ∩ Ω1, 0 on ∂Ω2 ∩ ∂Ω, and this procedure can be combined with numerical solution by, e.g., finite elements. The following alternative approach may be pursued when Ω1 and Ω2 are disjoint but with a common interface ∂Ω1 ∩ ∂Ω2: If uj denotes the solution in Ωj , j = 1, 2, then the transmission conditions u1 = u2, ∂u1/∂n = ∂u2/∂n have to be satisfied on the interface. One method is then to reduce the prob- lem to an integral type equation on the interface and use this as a basis of an iterative method. Bibliography Partial Differential Equations R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1–6, Springer-Verlag, Berlin, 1988–1993. L. C. Evans, Partial Differential Equations, American Mathematical Society, Prov- idence, RI, 1998. G. B. Folland, Introduction to Partial Differential Equations, second ed., Princeton University Press, Princeton, NJ, 1995. A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969. P. R. Garabedian, Partial Differential Equations, AMS Chelsea Publishing, Prov- idence, RI, 1998, Reprint of the 1964 original. F. John, Partial Differential Equations, fourth ed., Springer-Verlag, New York, 1991. I. G. Petrovsky, Lectures on Partial Differential Equations, Dover Publications Inc., New York, 1991, Translated from the Russian by A. Shenitzer, Reprint of the 1964 English translation. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equa- tions, Springer-Verlag, New York, 1984, Corrected reprint of the 1967 original. J. Rauch, Partial Differential Equations, Springer-Verlag, New York, 1991. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 1993. Functional Analysis L. Debnath and P. Mikusiński, Introduction to Hilbert Spaces with Applications, second ed., Academic Press Inc., San Diego, CA, 1999. E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York, 1989. W. Rudin, Functional Analysis, second ed., McGraw-Hill Inc., New York, 1991. G. F. Simmons, Introduction to Topology and Modern Analysis, Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1983. 254 Bibliography Finite Element Methods D. Braess, Finite Elements, second ed., Cambridge University Press, Cambridge, 2001. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Meth- ods, second ed., Springer-Verlag, New York, 2002. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive meth- ods for differential equations, Acta Numerica, 1995, Cambridge Univ. Press, Cam- bridge, 1995, pp. 105–158. G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer- Verlag, Berlin, 1997. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method. Vol. 1–3, Fifth edition, Butterworth-Heinemann, Oxford, 2000. Finite Difference Methods G. E. Forsythe and W. R. Wasow, Finite-Difference Methods for Partial Differen- tial Equations, John Wiley & Sons Inc., New York, 1960. B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time Dependent Problems and Differ- ence Methods, John Wiley & Sons Inc., New York, 1995. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1989. Other Classes of Numerical Methods K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997. J. P. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover Publica- tions Inc., Mineola, NY, 2001. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. G. Chen and J. Zhou, Boundary Element Methods, Computational Mathematics and Applications, Academic Press, London, 1992. J. Douglas, Jr. and T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathemat- ics, Vol. 385. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. Bibliography 255 R. Li, Z. Chen, and W. Wu, Generalized Difference Methods for Differential Equa- tions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 226, Marcel Dekker Inc., New York, 2000, A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equa- tions, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. W. L. Wendland, Boundary element methods for elliptic problems, Mathematical Theory of Finite and Boundary Element Methods (A. H. Schatz, V. Thomée, and W. L. Wendland, eds.), Birkhäuser Verlag, Basel, 1990, pp. 219–276. Numerical Linear Algebra J. H. Bramble, Multigrid Methods, Longman Scientific & Technical, Harlow, 1993. J. W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Ap- plied Mathematics (SIAM), Philadelphia, PA, 1997. P. Deuflhard and A. Hohmann, Numerical Analysis in Modern Scientific Comput- ing, second ed., Springer, New York, 2003. G. H. Golub and C. F. Van Loan, Matrix Computations, third ed., Johns Hopkins University Press, Baltimore, MD, 1996. A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, New York, 1999. B. F. Smith, P. E. Bjørstad, and W. D. Gropp, Domain Decomposition, Cambridge University Press, Cambridge, 1996. L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. R. S. Varga, Matrix Iterative Analysis, expanded ed., Springer-Verlag, Berlin, 2000. Index a posteriori error estimate 66 a priori error estimate 66 accurate of order r 135, 188 adjoint 55, 75 affine function 52 artificial diffusion 190, 209 assembly 67 Babuška-Brezzi inf-sup condition 72 backward Euler method 102, 140, 156 backward heat equation 112 Banach space 226 barycentric quadrature 68 basis function 52, 58 bilinear form 225 Biot number 10 boundary approximation 62 boundary element method 222 boundary integral method 221 bounded bilinear form 22, 228 bounded linear form 22 bounded linear operator 227 Bramble-Hilbert lemma 61, 75 C(M ), Cb(M ) 231 C(Rd) 231 Ck(Ω), Ck(Ω̄) 232 Ck0 (Ω) 232 Cauchy problem 2, 109 Cauchy sequence 226 Cauchy-Riemann equations 181 Cauchy-Schwarz inequality 226 CFL condition 190 characteristic boundary 170 characteristic curve 169 characteristic direction 163 characteristic polynomial 132, 163 characteristic surface 163 classical solution 21, 26 coercive bilinear form 21, 228 collocation method 217 compact set 6, 85, 231 compact support 232 complete space 226 conditional stability 158 conforming finite element method 71 conservation law 7 consistent 137 constitutive relations 8 convection-diffusion equation 12 convergent sequence 226 convolution 239 Courant-Friedrichs-Lewy condition 190 Crank-Nicolson method 103, 142, 158 curved boundary 62 cylindrical symmetry 13 d’Alembert’s formula 169 dense subspace 82, 233, 235 density argument 236–238 diffusion equation 12 dimensionless form 9 Dirac delta 23, 30, 241 Dirichlet’s boundary condition 9, 25 Dirichlet’s principle 22, 34 discontinuous Galerkin method 212 discrete Fourier transform 133 discrete Laplacian 151 discrete maximum-norm 45, 130 distribution 241 divergence 5 divergence form 10 divergence theorem 5 domain 232 258 Index domain of dependence 167, 177, 180, 190 dual space 227, 238 duality argument 55, 64, 66, 75 Dufort-Frankel scheme 137 Duhamel’s principle 118 elastic bar 12 elastic beam 13 elliptic equation 165 elliptic projection 64 energy estimate 229 energy norm 229 equivalent norms 227 essential boundary condition 36 Euler’s method 100 family of triangulations 60 Fick’s law 12 finite volume difference method 220 finite volume element method 220 finite volume method 219 finite-dimensional system of equations 231 forward Euler method 101, 130, 158 Fourier transform 109, 238 Fourier’s law 8 Friedrichs scheme 189, 192 Friedrichs system 178, 193 Friedrichs’ inequality 39 fundamental solution 30, 110 for Poisson’s equation 31 Galerkin’s method 53, 74 Gauss kernel 110 Gauss-Seidel method 247 generalized derivative 234 generalized function 241 gradient 5 Green’s formula 5 Green’s function 18, 23, 32, 73, 94, 127 Gronwall’s lemma 107, 179 harmonic function 11, 26, 28 heat equation 8 Hilbert space 226 Hk(Ω) 234 H10 (Ω) 238 H−1(Ω) 238 Hooke’s law 12 hyperbolic equation 165 hyperbolic system Friedrichs system 178, 193 strictly 174 symmetric 178, 193 inf-sup condition 72 inflow boundary 170 initial value problem 2 initial-boundary value problem 2 inner product 225 interpolation error 54, 61 interpolation near the boundary 49 interpolation operator 54, 60 inverse inequality 92, 94, 148, 158, 160 Jacobi method 247 Laplace operator 5 Laplace’s equation 10, 26 largest eigenvalue 92 Lax-Milgram lemma 22, 229 Lax-Wendroff scheme 191, 192 Lebesgue integral 232 linear form 225 linear functional 225 load vector 53, 58 Lp(Ω) 233 Lp-norm 233 L2(Ω) 233 L2(Γ ) 236 l2,h 133 l0h 141 l02,h 143 L2-norm 233 L2-projection 62 lumped mass method 153 mass matrix 73 maximum principle 16, 26, 122 discrete 44, 147, 154 strong 16, 18, 29 maximum-norm 6, 231 discrete 45, 130, 139 Maxwell’s equations 183 method of characteristics 171 min-max principle 84 minimum principle 16 Index 259 monotonicity property 18 multi-index 5 natural boundary condition 36 Neumann problem 35 Neumann’s boundary condition 9, 26 nodal quadrature 69, 153 non-conforming finite element method 71 nonlinear equations 11 norm 226 of operator 227 scaling of 242 normal derivative 5 operator norm 227 order of accuracy 135 orthogonal projection 227 orthonormal basis 81, 114, 166 outflow boundary 170 parabolic boundary 122 parabolic equation 165 Parseval’s formula 239 Parseval’s relation 83 Peclet number 10 Petrov-Galerkin method 210 Πk 56 Poincaré’s inequality 238 Poisson’s equation 10, 26 Poisson’s integral formula 28 pre-compact set 85 principal part 163 projection theorem 227 pseudospectral method 219 quadrature formula 68 quasi-optimal approximation 63 quasi-uniform family 65, 72, 92, 94, 153, 212 Raviart-Thomas element 72 regularity estimate 23, 37 relaxation 247 Rellich’s lemma 85 Riesz representation theorem 22, 34, 228 Ritz projection 64 Robin’s boundary condition 9, 26 R, R+ 5 scalar product 225 scaled trace inequality 67, 242 scaling 242 semidiscrete approximation 150 semigroup property 97, 125 seminorm 226, 232, 235 separation of variables 114 Shortley-Weller approximation 49 smooth function 6, 232 smoothing property 113 Sobolev imbedding 243 Sobolev inequality 237 Sobolev space 235 sparse matrix 58 spectral method 218 spherical symmetry 13 standard Galerkin method 208 stiff system 104 stiffness matrix 53, 58 Stokes equations 127 Strang’s first lemma 71 streamline 171 streamline diffusion method 209 strictly hyperbolic system 174 strong solution 21, 33 superconvergence 155, 218 symbol 132 symmetric hyperbolic system 178 θ-method 146 trace inequality 236 scaled 242 trace operator 235 trace theorem 236 triangulation 57 Tricomi’s equation 181 truncation error 45, 47, 131 unconditional stability 156 upwind scheme 187 variational equation 229 variational formulation 20, 33, 120 von Neumann condition 133, 134, 188, 194 wave equation 12 260 Index weak derivative 234 weak formulation 20, 33, 120 weak solution 21, 33 weakly imposed boundary condition 216 well posed problem 4, 112 Wendroff box scheme 196 W kp (Ω) 235 Z 186 |Ω| 5 Texts in Applied Mathematics (continued from page ii) 34. Chicone: Ordinary Differential Equations with Applications, Second Edition. 35. Kevorkian: Partial Differential Equations: Analytical Solution Techniques, Second Edition. 36. Dullerud/Paganini: A Course in Robust Control Theory: A Convex Approach. 37. Quarteroni/Sacco/Saleri: Numerical Mathematics. 38. Gallier: Geometric Methods and Applications: For Computer Science and Engineering. 39. Atkinson/Han: Theoretical Numerical Analysis: A Functional Analysis Framework, Second Edition. 40. Brauer/Castillo-Chávez: Mathematical Models in Population Biology and Epidemiology. 41. Davies: Integral Transforms and Their Applications, Third Edition. 42. Deuflhard/Bornemann: Scientific Computing with Ordinary Differential Equations. 43. Deuflhard/Hohmann: Numerical Analysis in Modern Scintific Computing: An Introduction, Second Edition. 44. Knabner/Angermann: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. 45. Larsson/Thomée: Partial Differential Equations with Numerical Methods. 46. Pedregal: Introduction to Optimization. 47. Ockendon/Ockendon: Waves and Compressible Flow. 48. Hinrichsen/Pritchard: Mathematical Systems Theory I. 49. Bullo/Lewis: Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems. 50. Verhulst: Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. 51. Bondeson/Rylander/Ingelström: Computational Electromagnetics. 52. Holmes: Introduction to Numerical Methods in Differential Equations. 53. Pavliotis/Stuart: Multiscale Methods: Averaging and Homogenization. 54. Hesthaven/Warburton: Nodal Discontinuous Galerkin Methods. 55. Allaire/Kaber: Numerical Linear Algebra. Lecture13p Thedeepness of freedom are threevalues at thenude functional Notconforming patrtaf.us vi sci x I beease ittouch 41 u VCsci inhalfedgeL U VCI't x Since u CPz are sci sc 7 that it have 3 Zeusunless e o E is P anisolvent it forgiven I lop C P s t 4 p di same degree y i l N Yi C E Sabi n ofsystem YCp g This is equivalent to say theonlypolynomial C PthetinterpolateZero data Yifp o is the Zeno poly vcpi.POTFF.gg In Edem e e I CRIvalue VCR Ca Ya metfunctor p E3 pjJ Chip J Shun E is p unisolvent Y Cul VCR7 0 Xz V UCR o rf VI UCB 0 Then over the edge PP we hone C P havingtworootsPR D This implies we 0 If e consider the other to edges G e b thesame argument we can see Eo tht means W Lv o hersonly trivial Solution Then Yi CUI Ri for any Xi E is P unisowent y csiy Ya f P Y cnn.PT III Ldj Pg I Pre 2 ily a PyO ein a 451214 7 f p i y g d CP f ftp b f CRI I B so fickle Cps O y Cp 7 L Escaple5 in lectureto Lecture_03_S08 LECTURE # 3: ABSTRACT RITZ-GALERKIN METHOD MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV 1. Variational Formulation In the previous lecture we have introduced the following space of functions defined on (0, 1): (1) V =  v : v(x) is continuous function on (0, 1); v′(x) exists in generalized sense and in L2(0, 1); v(0) = v(1) = 0   := H10 (0, 1) and equipped it with the L2 and H1 norms ‖v‖ = (v,v)1/2 and ‖v‖V = (v,v) 1/2 V = (∫ 1 0 (u′2 + u2)dx )1 2 . We also introduced the following variational and minimization problems: (V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V, (M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V, where a(u,v) is a bilinear form that is symmetric, coercive and contin- uous on V and L(v) is continuous on V and F(v) = 1 2 a(u,u) −L(v). As an example we can take a(u,v) ≡ ∫ 1 0 (k(x)u′v′ + q(x)uv) dx and L(v) ≡ ∫ 1 0 f(x)v dx. Here we have assumed that there are positive constants k0, k1, M such that (2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).

These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).

The proof of these properties follows from the following theorem:

Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:

(3)
|u(x)|2 ≤ C1

∫ 1
0

(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0

∫ 1
0

(u′(x))2dx.

with constants C0 and C1 that are independent of u.
1

2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV

Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.

Indeed, for any x ∈ (0, 1) we have:

u(x) = u(0) +
∫ x

0
u′(s)ds.

Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:

(4) |u(x)|2 =
∣∣∣∫ x

0
u′(s)ds

∣∣∣2 ≤ ∫ x
0

1ds
∫ x

0
(u′(s))2ds ≤ x

∫ x
0

(u′(s))2ds.

Taking the maximal value of x on the right hand side of this inequality
we get the first inequality (3) with C1 = 1. Further, increasing the r.h.s by
taking the integral in the whole interval and then integrating the obtained
inequality for x ∈ (0, 1) we get the second inequality (3) with C0 = 1/2.

This simple proof uses only one boundary condition u(0) = 0. We can
improve the constants if we use also the second boundary condition u(1) = 0.
Namely, we derive in the same manner the inequality

(5) |u(x)|2 = |
∫ 1
x
u′(s)ds |2 ≤

∫ 1
x

1ds
∫ 1
x

(u′(s))2 ≤ (1 −x)
∫ 1
x

(u′(s))2ds.

Now one multiplies (4) by 1 −x and (5) by x and adds the two inequalities
to get the estimate:

|u(x)|2 ≤ x(1 −x)
∫ 1

0
(u′(s))2ds.

This will allows us to show (3) with C0 = 1/6 and C1 = 1/4, correspondingly.
The appropriate inequalities for an arbitrary l are obtained by change of

the variable.
Now the coercivity of the bilinear form follows easily from these inequal-

ities and the assumptions (2). Indeed,

a(u,u) ≥ k0
∫ 1

0
u′2dx ≥

k0
2

∫ 1
0

(u′2 + u2)dx ≥
k0
2
‖u‖2V .

2. Abstract form of Ritz-Galerkin method

Instead of (V ), we shall consider its approximation. Namely, let Vh be a
n-dimensional subspace of V . We consider the following simpler problem:

(Vh) find uh ∈ Vh such that a(uh,v) = L(v), ∀ v ∈ Vh.
One can show that this problem is equivalent to the following minimization
problem in Vh:

(Mh) find uh ∈ Vh such that F(uh) ≤ F(v), ∀ v ∈ Vh.
There some simple but important for the applications properties that

are easily obtaind from the equivalence of the problems (Vh) and (Mh).
Using the fact that a(u,u) = L(u) and a(uh,uh) = L(uh) we get from the
inequality F(u) ≤ F(uh) that

1
2
a(u,u) −L(u) ≤

1
2
a(uh,uh) −L(uh) =⇒ −

1
2
a(u,u) ≤−

1
2
a(uh,uh)

LECTURE # 3: ABSTRACT RITZ-GALERKIN METHOD 3

which gives a(u,u) ≥ a(uh,uh). This inequality has a clear physical inter-
pretation.

Since Vh is n-dimensional, we can assume that it is spanned by n linearly
independent functions φj(x) ∈ V, j = 1, …,n; i.e.

Vh =


v : v(x) =

n∑
j=1

cjφj(x), cj are arbitrary constants


 .

We may relate the parameter h to n by h = 1/n.

Examples of Spaces Vh:

Example 1:

φj(x) = sin(jπx), j = 1, …,n. This will produce the so-called spectral
method.

Example 2:

φj(x) = xj(1 −x), j = 1, …,p. The so-called p-version of FEM.

Example 3:

The construction of the space is done in the following manner: split the
interval [0, 1] into n + 1 subintervals by introducing the points xj = jh,
j = 0, …,n + 1, where h = 1

n+1
. The space Vh consists of all continuous

functions on [0, 1] that are linear on each subinterval (element) [xj−1,xj]
and vanish at x = 0 and x = 1. Obviously, the functions in the space Vh are
determined by their values at the nodes xj, j = 1, …,n. Solving (Vh) with
such space Vh will lead to the finite element method.

The following set of functions can serve as a basis for Vh:

(6) φj(x) =




x−xj−1
h

, x ∈ [xj−1,xj];
xj+1 −x

h
, x ∈ [xj,xj+1];

0, elsewhere ;

j = 1, …,n. The functions are constructed in such way that φj(x) is 1 at
the node xj, 0 at all remaining nodes, and linear over the finite elements.
This basis is called a nodal basis. Note, that this is just one possible basis.
Another example is the so-called hierarchical basis.

Obviously, the solution uh of the problem (Vh) is in the form

uh(x) =
n∑
j=1

ξjφj(x), where ξj are unknown constants.

4 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV

0 1x

j

j

φ

Figure 1. A nodal basis function for linear finite elements

Then the method (Vh) can be written in the form

a(uh,v) = l(v), ∀v ∈ Vh =⇒ a


 n∑
j=1

ξjφj(x),φk


 = L(φk), k = 1, …,n.

(7)

This produces a linear system called also
(Ritz or Galerkin system) for the unknown ξ ∈ Rn:

n∑
j=1

ξja(φj,φk) = L(φk), k = 1, …,n, in matrix form Aξ = b,

where A ≡{ajk}nj,k=1 = {a(φj,φk)}
n
j,k=1, is a square n×n matrix

and b = {L(φj)}nj=1, and ξ = {ξj}
n
j=1 are vector-columns in R

n.

The matrix A is often called “stiffness” matrix while b is the “load” vector
which is computed from the data. Since the bilinear form a(., .) is coercive
the matrix A is nonsingular (show this) and therefore the system Aξ = b
has unique solution for any b. However, the condition number of A play an
importnat role in the numerical methods for solving the system and there
is a necessity to discuss this in details.

3. Mixed boundary conditions

For the boundary value problem

(D)
−(k(x)u′)′ + q(x)u = f(x), in (0, 1)

u(0) = 0,
u(1) + k(1)u′(1) = β1

we introduced the space V :

(8) V =


v :

v(x) is continuous function on (0, 1);
v′(x) exists in a generalized sense and is in L2(0, 1);
v(0) = 0




and the variational formulation (V ) with

a(u,v) ≡
∫ 1

0
(k(x)u′v′ + q(x)uv) dx + u(1)v(1)

LECTURE # 3: ABSTRACT RITZ-GALERKIN METHOD 5

and

L(v) ≡
∫ 1

0
f(x)v dx + β1v(1).

Examples of Spaces Vh for the problem (D):

Example 1:

φj(x) = sin((j − 0.5)πx), j = 1, …,n. This will produce the so-called
spectral method.

Example 2:

φj(x) = xj, j = 1, …,p. This will produce the so-called p-version of
Galerkin method.

Example 3: (the finite element method)

The construction of the space is done in the following manner: split the
interval [0, 1] into n subintervals by introducing the points xj = jh, j =
0, …,n, where h = 1

n
. The space Vh consists of all continuous functions on

[0, 1] that are linear on each subinterval (element) [xj−1,xj] and vanish at
x = 0. Obviously, the space Vh is determined by the values of a function at
the nodes xj, j = 1, …,n. Solving (Vh) with such space Vh will lead to the
finite element method.

In this case, a nodal basis will consist of all functions from the previous
example corresponding to internal nodes as well as one more function that
will involve the value at end xn = 1: The following set of functions can serve
as a basis for Vh:

(9) φn(x) =




x−xn−1
h

, x ∈ [xn−1,xn];

0, elsewhere.

Another possible basis in Vh is the so-called hierarchical basis which utilizes
hierarchy of grids.

4. Neumann boundary conditions

Now we shall consider the following simple model problem for the un-
known function u(x):

(D)
−u′′ + u = f(x), in (0, 1)

u′(0) = 0.
u′(1) = 0.

In the previous lecture we have introduced the set of functions defined on
(0, 1) that are is continuous function, have piece-wise continuous deriva-
tive.This set has been equipped with the norms

||v||2 = (v,v) and ||v||2V = (v,v)V = (v,v) + (v
′,v′).

6 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV

After completing the set V in the norm || · ||V we get the Sobolev space
H1(0, 1) of functions having generalized first derivatives in L2(0, 1). Note,
that the functions in V do not satisfy any boundary conditions. Therefore,
V ≡ H1(0, 1).

We also introduced the following variational problem:

(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
where

a(u,v) ≡
∫ 1

0
(u′v′ + uv) dx and L(v) ≡

∫ 1
0
f(x)v dx.

We shall study the Ritz system for this particular BVP. It is obvious, that
a(u,v) = (u,v)V so this form is trivially coercive.

We have introduced the following finite dimensional space: split the in-
terval [0, 1] into n− 1 subintervals by introducing the points xj = (j − 1)h,
j = 1, …,n, where h = 1

n−1 ; the space Vh consists of all continuous on [0, 1]
functions that are linear on each subinterval (element) [xj−1,xj]. Obviously,
the functions in the space Vh can be determined by their values at the nodes
xj, j = 1, …,n. The approximate problem (Vh) for such space Vh will lead
to the finite element method with linear elements.

The following set of functions can serve as a basis for Vh:

(10) φj(x) =




x−xj−1
h

, x ∈ [xj−1,xj];
xj+1 −x

h
, x ∈ [xj,xj+1];

0, elsewhere ;

j = 2, …,n− 1 and two additional functions defined at the end-points:
(11)

φ1(x) =




x2 −x
h

, x ∈ [x1,x2];

0, elsewhere ;
φn(x) =




x−xn−1
h

, x ∈ [xn−1,xn];

0, elsewhere .

The functions are constructed in such way that φj(x) is 1 at the node xj,
0 at all remaining nodes and linear over the finite elements. This basis is
called nodal basis.

The solution uh of the problem (Vh) is in the form

uh(x) =
n∑
j=1

ξjφj(x), where ξj are unknown constants.

Then the method (Vh) can be written in the form (7).
This basis of the space Vh will produce a tridiagonal matrix A. Indeed,

a(φj,φk) = 0, for |j −k| > 1. Also, for j = k we get

a(φj,φj) =
∫ xj+1
xj−1

(
1
h2

+ φ2j (x)
)
dx =

2
h

+
2h
3
, for 1 < j < n, a(φ1,φ1) = ∫ x2 x1 ( 1 h2 + φ21(x) ) dx = 1 h + h 3 , for j = 1, a(φn,φn) = ∫ xn xn−1 ( 1 h2 + φ2n(x) ) dx = 1 h + h 3 , for j = n. LECTURE # 3: ABSTRACT RITZ-GALERKIN METHOD 7 Similarly, for k = j + 1 we get a(φj,φj+1) = ∫ xj+1 xj ( −1 h2 + φj(x)φj+1(x) ) dx = −1 h + h 6 . The coefficients below the main diagonal are recover from the symmetry of the matrix A. Thus, the matrix A of the Ritz-system has the form A = A0 + A1, where: (12) A1 = 1 h   1 −1 0 . . . 0 −1 2 −1 . . . 0 0 −1 2 . . . 0 . . . . . . . 0 0 0 . . . 1   , A0 = h6   2 1 0 . . . 0 1 4 1 . . . 0 0 1 4 . . . 0 . . . . . . . 0 0 0 . . . 2   . Matrix A1 is called “stiffness” matrix, while the matrix A0 is called “mass” matrix. Both matrices are symmetric and A0 is positive definite while A1 is semi-definite. 5. Issues to be addressed In the genral case we are facing the following issues: • to assemble the matrix A and to solve the system Aξ = b; • alternatively, if an iterative method is used that requires only the matrix-vector multiplication Aξ, then one should prepare a pro- cedure of matrix vector multiplication (possibly without explicitly forming the matrix A); • estimate the condition number of the matrix A for a prticular chice of the basis of the space Vh; • estimate the error e = u−uh; • to develope an algorithm that adaptively choses the mesh so that the error is uniformly distributed in the domain and is dreven below a desired level. We need to develop the mathematical tools for studying these problems. This includes: estimate for the condition number of A, deriving/finding fast methods for solving the system, proving various integral inequalities, deriv- ing the approximation error with piece-wise polynomial functions, estimates in various Sobolev norms, etc. 6. An estimate of the condition number of the global matrix A for Neumann BC Further, we shall use the following definition of a condition number of a symmetric and positive definite matrix: cond(A) = max λ(A) min λ(A) , where λ(A) is an eigenvalue of A, i.e. Aξ = λξ, for some ξ a nonzero vector in Rn. Often it is not possible to compute the eigenvalues and the condition number, but for practical purposes it is enough to have an upper bound for cond(A). For this we need upper and lower bounds for the eigenvalues of A. 8 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV Simple calculations show that h 6 ≤ λ(A0) ≤ h and 0 ≤ λ(A1) ≤ 4 h . So we produce the following bound from above for the condition number of the matrix A (13) cond(A) ≤ max λ(A0) + max λ(A1) min λ(A0) + min λ(A1) ≤ 4/h + h h/6 = 24 h2 + 6 = O(h−2). Remark 1. Note, that A0 and A1 are square matrices of size n and one finds that cond(A0) ≤ 6 i.e. the condition number of A0 does not depend on the size of the matrix. Such matrices are called well-conditioned. In contrast, A1 has condition number O(h−2) which increases quadratically, when h → 0. Such matrices are called ill-conditioned. 7. Exercises The following matrices play essential role in the finite element, finite vol- ume and finite difference methods for two-point boundary value problems and the solution of the corresponding linear systems. The spectral properties of these matrices are used very often in the computational practice. (1) Find the exact eigenvalues of the matrices B1,B0 ∈ Rn×n given by (14) B1 =   2 −1 0 . . . 0 0 −1 2 −1 . . . 0 0 0 −1 2 . . . 0 0 . . . . . . . . 0 0 0 . . . −1 2   , and (15) B0 =   4 1 0 . . . 0 0 1 4 1 . . . 0 0 0 1 4 . . . 0 0 . . . . . . . . 0 0 0 . . . 1 4   . Hint: Show that λj(B1) = 4 sin2 πj 2(n+1) , j = 1, . . . ,n and then use the fact that B1 +B0 = I, where I is the identity matrix in Rn. From these calculations follow that both B1 and B0 are positive definite. (2) Estimate that eigenvalies of the scaled “stiffness” matrix B1 ∈ Rn×n (16) B1 =   1 −1 0 . . . 0 0 −1 2 −1 . . . 0 0 0 −1 2 . . . 0 0 . . . . . . . . 0 0 0 . . . −1 1   , and the scaled “mass” matrix B0 ∈ Rn×n (17) B0 =   2 1 0 . . . 0 0 1 4 1 . . . 0 0 0 1 4 . . . 0 0 . . . . . . . . 0 0 0 . . . −1 2   . LECTURE # 3: ABSTRACT RITZ-GALERKIN METHOD 9 Remark 2. Using the technique applied above we can show that in this case the eigenvalues are λj(B1) = 4 sin2 πj 2(n−1) , j = 0, . . . ,n− 1. Remark 3. The eigenvalues and eigenvectors of these algebraic problems and problems obtained by approximation of the same differential operator with third type boundary conditions could be found in the monograph of Samarskii [6, pp. 104–109]. References [1] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, AMS, 1998. [2] Ch. Grossmann, H.-O. Ross, and M. Stynes, Numerical Treatment of Partial Differ- ential Equations, Springer, Berlin, 2005. [3] M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics, Springer-Verlag, 1993. [4] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic PDEs, Springer-Verlag, New Yrok Inc, 2003. [5] S. Larsen and V. Thomee, Partial Differential Equations with Numerical Methods, Springer, 2003. [6] A.A. Samarskii, The Theory of Difference Schemes, Monographs and Textbooks in Pure and Appled Mathematics, Marcel Dekker, Inc, New York, 2001. Lecture_08_S08 LECTURE # 8: MULTIDIMENSIONAL SECOND ORDER ELLIPTIC PROBLEMS MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV 1. Introduction and preliminaries First, we introduce some notations that will be used further. Here Ω will denote a polygonal bounded domain in Rd, d = 2, 3 with boundary ∂Ω. Further, for the vector q = (q1, . . . , qd) and for a scalar function v we define the divergence ∇ · q and the gradient ∇v, correspondingly, by ∇ · q = ∂q1 ∂x1 + · · · + ∂qd ∂xd and ∇v = ( ∂v ∂x1 , . . . , ∂ ∂xd ) . The Stokes theorem will be used in the following form: ∫ ∂Ω q · n ds = ∫ Ω ∇ · q dx. Here, n is the outward unit vector to ∂Ω and q·n denotes the inner product of two vectors on Rd. We shall use the Hilbert space H1(Ω) of functions defined on Ω and having their generalized derivatives in L2(Ω). The subspace of those functions in H1(Ω) that vanish on the boundary ∂Ω will be denoted by H10 (Ω). The L2 and H1-inner products of these spaces and the corresponding norms are defined as follows: (u, v) = ∫ Ω uv dx, (u, v)1 = (u, v) + (∇u, ∇v),(1) ‖u‖ = (u, u)1/2, ‖u‖1 = (u, u)1/21 .(2) For the elements in the space H1(Ω) we shall use the following Poincare inequality: (3) ∫ Ω u2 dx ≤ M0 ∫ Ω |∇u|2 dx where the constant M0 > 0 does not depend on u.
We shall give proof of this inequality for d = 2.Without loss of generality,

we can assume that Ω is contained in the unit square Π, i.e. Ω ⊂ Π :=
(0, 1) × (0, 1). Then we can extend a function u ∈ H1(Ω) to Π by zero

1

2 MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

outside Ω. The extended function is denoted by ū. It belongs to H10 (Π) and
obviously,

∫

Π
ū2 dx =

∫

Ω
u2 dx.

Next, we write the equality

2
∫

Π
ū2 dx =

∫

Π

{(∫ x1
0

∂

∂x1
ū(ξ, x2) dξ

)2
dx(4)

+
(∫ x2

0

∂

∂x2
ū(x1, ξ) dξ

)2 }
dx

and apply Cauchy-Schwarz inequality to each of the line integrals:

2
∫

Π
ū2 dx ≤

∫

Π

{
x1

∫ 1
0

(
∂

∂x1
ū(ξ, x2) dξ

)2
(5)

+x2
∫ 1

0

(
∂

∂x2
ū(x1, ξ) dξ

)2 }
dx.

Using Fubini theorem,we get finally:

2
∫

Ω
u2 dx = 2

∫

Π
ū2 dx ≤ 1

2

∫

Π
|∇ū|2 dx(6)

=
1
2

∫

Ω
|∇u|2 dx,

which is the required inequality with M0 = 1/4. If the domain Ω is contained
in a rectangle (0, l1)×(0, l2) the required inequality follows by change of the
variables.

Further, we shall need the following two inequalities valid for functions in
H1(Ω):

(7)
∫

∂Ω
u2 ds ≤ C‖u‖21,

and

(8)
∫

Ω
u2 dx ≤ C

{∫

Ω
|∇u|2 ds +

∫

∂Ω
u2 ds

}
.

Here the constant C does not depend on u but depend on the domain Ω.
One can prove these inequalities for rectangular domains simply by using
the corresponding estimates from the one-dimensional case. The proofs are
left as an exercise for this part of the class (see, e.g. [3, 7]).

MULTIDIMENSIONAL ELLIPTIC PROBLEMS 3

2. Problem formulation

In this lecture we shall consider the following Dirichlet boundary-value
problem: find u(x) such that:

(D)
Lu := ∇ ·

(
−K(x)∇u + b(x)u

)
+ q(x)u = f (x), x ∈ Ω

u(x) = 0, x ∈ ∂Ω.

where the coefficients K(x), b, q and f are given functions on Ω. We
assume that Ω is a bounded domain with Lipschitz boundary ∂Ω, K(x) is a
symmetric and uniformly in Ω positive definite matrix and the coefficients
K(x), b(x), q(x) are measurable and bounded function in Ω.

This is the divergent form of the problem. Quite often second order
problems are given in the following non-divergent form:

(9)
Lu := ∇ · (−K(x)∇u) + b̃(x)∇u + q(x)u = f (x), x ∈ Ω

u(x) = 0, x ∈ ∂Ω.

If the vector field b̃(x) is differentiable then these two forms are equivalent.
In case when b ≡ b̃ and ∇ · b = 0, then these two form coinside.

In some applications this equation describes: (1) deflection of an elastic
membrane under transverse load f (then K = I, b ≡ 0, q ≡ 0); (2) the
pressure distribution in a porous media (K is the permeability tensor, b ≡
0, q ≡ 0); (3) concentration distribution of a chemical in a flow with velocity
b and absorption coefficient q. The quantity

q(x) = −K(x)∇u + b(x)u
is often called total flux (mass, thermal, etc) with −K(x)∇u the diffusive
part and b(x)u convective part of the flux.

For deriving the variational formulation of this problem we follow the
standard approach used in the 1-dimensional problems. We multiply the
differential equation (D) by a test function v ∈ H10 (Ω) and integrate over Ω:

∫

Ω

(
∇ · (−K(x)∇u + b(x)u) + q(x)u

)
v dx =

∫

Ω
f (x)v dx.

We use the identity
(
∇ · (−K(x)∇u + b(x)u)} v = ∇ · {(−K(x)∇u + b(x)u) v

)
(10)

−
(
− K(x)∇u + b(x)u

)
· ∇v,

4 MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

so that after applying the Stokes theorem we transform the right hand side
of the above identity to the form:

∫

∂Ω

(
K(x)∇u − b(x)u

)
· n v ds +

∫

Ω

(
K(x)∇u − b(x)u

)
· ∇v dx.

Now we use the fact that v vanishes on ∂Ω to get
∫

Ω

(
(K(x)∇u − b(x)u) · ∇v + q(x)uv

)
dx =

∫

Ω
f (x)v dx.

We rewrite this integral identity in the abstract form

a(u, v) = L(v) ∀ v ∈ H10 (Ω),
where

a(u, v) =
∫

Ω

(
K(x)∇u · ∇v − ub(x) · ∇v + q(x)uv

)
dx

and

L(v) =
∫

Ω
f (x)v dx.

Thus, we have shown that the solution of the problem (D) satisfies the
following variational problem:

(V ) find u ∈ H10 (Ω) such that a(u, v) = L(v), ∀ v ∈ H10 (Ω) .

So we have reformulated the differential problem (D) in terms of integral
identity involving the bilinear form a(·, ·) and the linear form L(·). Again,
we can use the general theoretical framework and Lax-Milgram theorem to
show the existence and the uniqueness of the solution u ∈ H10 (Ω). We shall
prove that under reasonable conditions on the coefficients the bilinear form
a(·, ·) is coercive and continuous in V = H10 (Ω) so we can apply the general
theoretical framework for such problems.

Now we give conditions on the coefficients of the differential equation (D)
that are sufficient for the coercivity and the continuity of the bilinear form
a(·, ·):

(C)
ξT K(x)ξ ≥ k0ξT ξ, ∀ ξ ∈ Rd, k0 = const > 0,

q(x) + 1
2
∇ · b(x) ≥ 0, ∀ x ∈ Ω.

Theorem 1. Assume that the conditions (C) are satisfied. Then the bilinear
form a(·, ·) is coercive and continuous in V , i.e. there are positive constants
α and C such that

(11)
a(u, u) ≥ α‖u‖21, (coercivity)
a(u, v) ≤ C0‖u‖1 ‖v‖1. (continuity)

MULTIDIMENSIONAL ELLIPTIC PROBLEMS 5

Proof: First, we note that

(12) −u b · ∇v = −1
2
∇ · (bu2) + 1

2
u2 ∇ · b.

Then applying the Stokes theorem and the condition (C) and the fact that
v vanishes on ∂Ω we get the following for for a(u, u)

a(u, u) =
∫

Ω
(K∇u · ∇u + (q + 0.5∇ · b)u2) dx ≥ k0||∇u||2.

Using Poincare inequality (3) we get the desired result regarding the coer-
civity. The continuity of the bilinear from is a simple consequence of the
boundness of the coefficients.

Let Vh be a finite dimensional subspace H10 (Ω). The Ritz-Galerkin method
can be formulated in the already discussed abstract form:

(Vh) find uh ∈ Vh ⊂ H10 (Ω) such that a(uh, v) = L(v), ∀ v ∈ Vh.

Our goal now is to construct the space Vh and to show how the Ritz-system
derived from (Vh) is computed and solved.

3. Other types of boundary conditions

Instead of Dirichlet boundary conditions one can put various other types
of boundary conditions on ∂Ω. Below we give two natural boundary condi-
tions that are widely used in the applications.

Case b ≡ 0; Then we have diffusion-reaction equation and the following
Robin condition can be prescribed on the whole boundary ∂Ω or on part of
it:

(13) K(x)∇u · n + σ(x)u = g(x) ∀ x ∈ ∂Ω.
Here σ(x) ≥ 0 and g(x) are given functions on ∂Ω. If σ(x) ≡ 0 then this is
the classical Neumann boundary condition. The meaning of this boundary
condition is that we prescribe the diffusive flux on ∂Ω. If σ(x) ≡ g(x) ≡ 0
then no flux is allowed through ∂Ω. This is typical insulated boundary (in
thermal problem) or no-flow boundary in porous media applications.

The weak formulation of this boundary-value problem is obtained in the
same way as in the case of Dirichlet boundary conditions. Then after inte-
gration by parts and using the boundary conditions we get that u ∈ H1(Ω)
satisfies the following integral identity:

a(u, v) = L(v) ∀ v ∈ H1(Ω),
where

a(u, v) =
∫

Ω
{K(x)∇u · ∇v + q(x)uv} dx +

∫

∂Ω
σu v ds

and
L(v) =

∫

Ω
f (x)v dx +

∫

∂Ω
g(x) v ds.

6 MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

Note, that the functions in the solution space doe not satisfy any boundary
conditions. The bilinear form is coercive in H1(Ω) under the condition that
σ(x) ≥ σ0 = const > 0 ∀ x ∈ ∂Ω. In fact, it is enough that σ(x) ≥ σ0 > 0
on a part of the boundary with a positive measure. Indeed, we have

a(u, u) ≥ k0
∫

Ω
|∇u|2 dx + σ0

∫

∂Ω
u2 ds.

Next, we use the embedding inequality (8) to get the missing ‖u‖2-term in
the coercivity.

Similarly,

|L(v)| ≤
∫

Ω
|f v| dx +

∫

∂Ω
|g v| ds(14)

≤‖f‖‖v‖ +
(∫

∂Ω
|g|2 ds

)1/2 (∫

∂Ω
|v|2 ds

)1/2
.(15)

Finally, we use the estimate (7) to get the required continuity of the linear
form L(v).

Case b 6≡ 0; This diffusion-convection-reaction equation and the fol-
lowing boundary conditions are quite natural (together with Dirichlet BC).
First, split the boundary ∂Ω into two parts: ∂Ω = Γin ∪ Γout, where

Γin = {x ∈ ∂Ω : b(x) · n(x) < 0}, Γout = {x ∈ ∂Ω : b(x) · n(x) ≥ 0}. Then the following boundary conditions are natural: (16) −K(x)∇u · n = 0, x ∈ Γout, −K(x)∇u · n + u b · n = g(x), x ∈ Γin. The physical meaning of these boundary conditions is the following: on the part of the boundary where the flow enter the domain, i.e. b(x) · n(x) < 0 we can prescribe either the function or the total flux q. Then one gets the following form (Lu, v) = ∫ Ω {K(x)∇u · ∇v − b · ∇vu + q(x)uv} dx(17) + ∫ Γout b · nu v ds + ∫ Γin g v ds Now we define the bilinear from a(·, ·) and the linear form L(·) as a(u, v) = ∫ Ω {K(x)∇u · ∇v − b · ∇vu + q(x)uv} dx + ∫ Γout b · nu v ds and L(v) = ∫ Ω f vdx + ∫ Γin g v ds. MULTIDIMENSIONAL ELLIPTIC PROBLEMS 7 Ωb Γ out Γ out Γ out Γ out Γ in Γ in Γ in Figure 1. Domain with inflow Γin and outflow Γout boundaries One can easily prove that if the coefficients of the differential equation satisfy one of the conditions (A) q(x) + 1 2 ∇ · b(x) ≥ c0 = const > 0 ∀ x ∈ Ω,

(B) q(x) +
1
2
∇ · b(x) ≥ 0 and the measure of the set Γin is nonzero,

(C) q(x) +
1
2
∇ · b(x) ≥ 0 and the measure of the set Γout is nonzero,

then the corresponding bilinear form will be coercive in H1(Ω)-norm.
Indeed, using (12) by Stokes’ theorem

a(u, u) =
∫

Ω
(K∇u · ∇u + (q + 0.5∇ · b)u2) dx(18)

− 1
2

∫

Γin
b · nu2ds + 1

2

∫

Γout
b · nu2ds.

which is the same as

a(u, u) =
∫

Ω
(K∇u · ∇u + (q + 0.5∇ · b)u2) dx + 1

2

∫

∂Ω
|b · n|u2ds.

Obviously one of the conditions (A) – (C) guarantee the coercivity of the
bilinear form in H1(Ω)-norm. Then, by Lax-Milgram Theorem, we get the
desired result about existence and uniqueness of the solution of the equation
(D) with boundary conditions (16).

For this problem the following maximum principle could be shown

Theorem 2. ([6, Theorem 31., p. 26]) Consider the differential operator L
of problem (D) and assume that u ∈ C2(Ω̄) and

Lu ≤ 0 (Lu ≥ 0) in Ω.

8 MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

(i) If q = 0, then

max
x∈Ω̄

u(x) ≤ max
x∈∂Ω

u(x)
(

min
x∈Ω̄

u(x) ≤ min
x∈∂Ω

u(x)
)

(ii) If q ≥ 0, then

max
x∈Ω̄

u(x) ≤ max(max
x∈∂Ω

u(x), 0)
(

min
x∈Ω̄

u(x) ≤ min( min
x∈∂Ω

u(x), 0)
)

This theorem allows us to study the uniqueness and the stability of the
solution of the problem (D) in maximum-norm. This is quite useful in many
applications. However, the natural way to numerically address this problem
is to use the variational form of the problem (D) and derive approximation
schemes generated by the finite element method.

4. Abstract Galerkin method

We take Vh is a finite dimensional subspace of V (denote the dimension
of Vh by n) and consider the variational problem on Vh:

(Vh) find uh ∈ Vh such that a(uh, v) = L(v), ∀ v ∈ Vh.
Let {φj (x)}ni=1 be a basis for Vh, so that for uh, v ∈ Vh we have:

(19) uh(x) =
n∑

i=1

Uiφi(x) and v(x) =
n∑

i=1

Viφi(x).

The parameters in the vector-column U T = (U1, . . . , Un1)T are called de-
grees of freedom for the finite element method and are obtained from the
Galerkin system AU = b. Here the entries of the matrix A are aij = a(φi, φj )
and the vector-load b has components bj = L(φj ). Since we have assumed
coercivity of the bilinear from a(·, ·), the matrix A is nonsingular and there-
fore the Galerkin system has unique solution.

Note that the matrix A is non-symmetric (as long as b 6= 0). Moreover,
for convection dominated problems that is when b is much larger than K (in
some norm) and this causes a number of serious problems for any numerical
method.

Our main goal in this class will construction of appropriate (and practi-
cally feasible) finite dimensional spaces Vh, the spaces of piece-wise polyno-
mial functions over a partition of Ω into finite elements. The basic texts we
shall use in this class are the textbooks of Larsen and Thomée [6] or [5] and
the monographs of Ciarlet [1], and Ern and Guermond [2].

MULTIDIMENSIONAL ELLIPTIC PROBLEMS 9

References

[1] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics of Applied
Mathematics, v. 40, SIAM, 2002.

[2] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Series of Applied
Mathematical Sciences v. 159, Springer-Verlag, 2004.

[3] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, volume
19, American Mathematical Society, 1991.

[4] D. Kinkaid and W. Cheney, Numerical Analysis. Mathematics of Scientific Comput-
ing, Third Edition, Brooks/Cole, 2002.

[5] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic PDEs,
Springer-Verlag, New Yrok Inc, 2003.

[6] S. Larsen and V. Thomée, Partial Differential Equations with Numerical Methods,
Springer-Verlag, Texts in Applied Mathematics 45, 2003.

[7] M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Texts
in Applied Mathematics 13, Springer-Verlag, 1993.

5. Appendix: Abstract variational problem:

We recall for the previous lectures the following general framework for
elliptic equations.

Let V be a Hilbert space with an inner product (·, ·)V and corresponding
norm || · ||V . Let the bilinear form a(u, v) defined on V D × V and the linear
form L(v) defined on V are such that:

(1) a(u, v) is coercive in V , i.e., there is a constant α > 0 such that
a(v, v) ≥ α||v||2V , ∀ v ∈ V ;

(2) a(u, v) is continuous, i.e., there is a constant C > 0 such that
a(u, v) ≤ C0||u||V ||v||V , ∀ u, v ∈ V ;

(3) L(v) is continuous in V , i.e., there is a constant Λ > 0 such that
L(v) ≤ Λ||v||V , ∀ v ∈ V .

The following theorem is a particular case of the well-know Lax-Milgram
theorem for NON-SYMMETRIC bilinear forms (see e.g. [7] for the proof
of the symmetric and your lecture notes, Lecture # 7, for non-symmetric
forms a(·, ·)):
Theorem 3. (Lax-Milgram) Let V be the Hilbert space with an inner product
(u, v)V and let the conditions (1) – (3) holds true. Then the problem find
u ∈ V s.t.
(20) a(u, v) = L(v), ∀ v ∈ V
has unique solution u ∈ V . Furthermore, the solution satisfies the stability
estimate:

‖u‖V ≤
Λ
α

.

Lecture_09_S08

LECTURE # 9:

INTRODUCTION TO FEM FOR SECOND ORDER
ELLIPTIC EQUATIONS

MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

As introduced before, Ω will be a polygonal bounded domain in Rd, d =
2, 3 with boundary ∂Ω. Further, for the vector q = (q1, . . . ,qd) and for
a scalar function v we define the divergence ∇ · q and the gradient ∇v,
correspondingly, by

∇· q =
∂q1
∂x1

+ · · · +
∂qd
∂xd

and ∇v =
(
∂v

∂x1
, . . . ,

∂

∂xd

)
.

No we consider the following model boundary-value problem:

(D)

find u(x) such that:

Lu := −∇·∇u + u := −∆u + u = f(x), x ∈ Ω

∇u · n :=
∂u

∂n
(x) = 0, x ∈ ∂Ω.

where a(u,v) =
∫

Ω
{∇u ·∇v + uv} dx and L(v) =

∫
Ω
f(x)v dx

The variational formulation of this problem is:

(V )
find u ∈ V := H1(Ω) such that a(u,v) = L(v), ∀ v ∈ H1(Ω),

where a(u,v) =
∫

Ω
{∇u ·∇v + uv} dx and L(v) =

∫
Ω
f(x)v dx.

So we have reformulated the differential problem (D) in terms of integral
identity involving the bilinear form a(·, ·) and the linear form L(·). Again,
we can use the general theoretical framework and Lax-Milgram theorem to
show the existence and the uniqueness of the solution u ∈ H1(Ω). Obviously,
the bilinear form a(·, ·) is coercive and continuous in V = H1(Ω) so we can
apply the general theoretical framework for such problems.

Date: September 20, 2010.

1

2 MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

Ω

τ
1

τ
2

τ
3

Figure 1. Left: Triangulation of a polygonal domain; Right:
non-conforming triangulation, which is not allowed in our
current considerations

Let Vh be a finite dimensional subspace H1(Ω). The Ritz-Galerkin method
can be formulated in the already discussed abstract form:

(Vh) find uh ∈ Vh ⊂ H1(Ω) such that a(uh,v) = L(v), ∀ v ∈ Vh.

Our goal now is to construct the space Vh and to show how the system of
linear equations derived from (Vh) is computed and solved.

We point out that the boundary conditions of the differential problem
are not imposed on the functions from the space H1(Ω). This condition is
weakly contained in the variational formulation itself. It is more natural to
begin with such a problem, since we do not need to impose any boundary
conditions on the finite dimensional subspace of H1(Ω).

1. FE partition of the domain and FE spaces

We partition the domain Ω into triangular (tetrahedral) finite elements τ.
The finite elements τ are considered open sets and we denote their closure
by τ, i.e. τ = τ ∪ ∂τ. This triangulation is denoted by Th. We shall
consider conforming types of triangulation, i.e. triangulations that satisfy
the following conditions:

(1)

(a) τ are disjoint, i.e. τi ∩ τj = ∅, i 6= j;

(b) τi ∩ τj is either:
(i) a vertex of τi & τj;
(ii) an entire edge of τi & τj;
(iii) empty .

An example of a triangulation of the domain is shown on Figure 1. The
right Figure 1 shows a non-conforming triangulation that is NOT considered
in our current setting.

FEM FOR ELLIPTIC PROBLEMS 3

P
i

Pj

P
k (Pk)=Uk

τ

u (P )=Uh
u (P )=Uh

uh

i jji

Figure 2. Liner triangular finite element

Together with set of all triangles Th we shall use the sets of all edges Eh
and the sets of all vertices Vh. Further, we define the set Pm of polynomials
of degree m with real coefficients:

Pm =


 ∑

0≤i+j≤m
cijx

i
1 x

j
2, cij are real numbers


 .

Now we consider the simpleast case of linear finite elements and define
the finite-dimensional subspace Vh ⊂ H1(Ω) in the following way:

Vh =
{
v : v ∈ C0(Ω), v ∈P1(τ), for τ ∈Th

}
.

The functions in Vh can be uniquely determined by its values at the
vertices of the triangulation Th. We shall use the nodal basis in Vh. If the
number of the vertices in Vh is N, then we define N linearly independent
functions φj(x), j = 1, . . . ,N by:

φj(x) =




1 if x = Pj, Pj ∈Vh;
0 if x = Pk, Pk ∈Vh, Pk 6= Pj;
linear over each τ ∈Th.

2. Finite Element Computations

Each function in Vh can be presented in the form:

uh(x) =
N∑
j=1

Ujφj(x), where Uj = uh(Pj), Pj ∈Vh.

Then the finite element method for the problem (V ) reduces to solving
the Ritz-Galerkin system of linear equations for the unknown values UT =
(U1,U2, . . . ,UN ):

(2)
N∑
i=1

Uia(φi,φj) = L(φj), j = 1, . . . ,N, or AU = b.

4 MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

The entries of the matrix A are a(φi,φj) and the entries of the load-vector
b are L(φj), i,j = 1, . . . ,N. Since the bilinear form a(·, ·) is symmetric and
coercive the matrix A is symmetric and positive definite.

The matrix A of the system (2) is computed element-wise. Namely, the
contributions of a particular finite element τ to the global “stiffness” and
“mass” matrices are done by element-wise computations.

In each element we introduce local notations: let the triangle τ has vertices
Pi, Pj, Pk and let the restrictions of the nodal basis functions to τ be
denoted again by φi, φj, φk. We denote

Uτ =


 UiUj
Uk


 , Vτ =


 ViVj
Vk


 , Φτ (x) := Φτ =


 φi(x)φj(x)
φk(x)


 ,

and similarly for the element functions

∇Φτ (x) := ∇Φτ =


 ∇φi∇φj
∇φk


 :=



∂φi
∂x1

∂φi
∂x2

∂φj
∂x1

∂φj
∂x2

∂φk
∂x1

∂φk
∂x2


 .

This allows us to write the following presentations:

uh(x)|τ = ΦTτ Uτ, v(x)|τ = Φ
T
τ Vτ, ∇uh(x)|τ = ∇Φ

T
τ Uτ, ∇v(x)|τ = ∇Φ

T
τ Vτ,

so that ∫
τ
∇uh(x) ·∇v(x) dx =

∫
τ
Vτ

T∇Φτ∇ΦTτ Uτ dx := V
T
τ A

1
τUτ,∫

τ
uh(x)v(x) dx :=

∫
τ
V Tτ Φτ Φ

T
τ Uτ dx := V

T
τ A

0
τUτ,

and similarly for the r.h.s.∫
τ
f(x)v(x) dx :=

∫
τ
V Tτ Φτf(x) dx := Vτ

Tbτ.

Here A1e and A
0
e are 3 × 3 matrices, called element “stiffness” and “mass”

matrices, correspondingly, and bτ is a vector of dimension 3, called the
element load vector.

3. Element Stiffness and mass matrices for linear FE

One gets a very simple formula for these matrices, namely:

A1τ =
∫
τ


 ∇φi ·∇φi ∇φi ·∇φj ∇φi ·∇φk∇φj ·∇φi ∇φj ·∇φj ∇φj ·∇φk
∇φk ·∇φi ∇φk ·∇φj ∇φk ·∇φk


dx,

FEM FOR ELLIPTIC PROBLEMS 5

8

9

i−1

y
j−1

y
j

y
j+1

τ1

τ
2

τ
3

τ
4

τ
5

τ
6

0 1

23

4

5 6

h h

h

h

x x
i

x
i+1

Figure 3. Uniform rectangular grid

and

A0τ =
∫
τ


 φiφi φiφj φiφkφjφi φjφj φjφk
φkφi φkφj φkφk


dx =



∫
τ
φiφi

∫
τ
φiφj

∫
τ
φiφk∫

τ
φjφi

∫
τ
φjφj

∫
τ
φjφk∫

τ
φkφi

∫
τ
φkφj

∫
τ
φkφk


 .

Further, we shall show that in fact the element “mass” matrix A0τ has very
simple form, namely:

A0τ =
|τ|
12


 2 1 11 2 1

1 1 2


 ,

where |τ| denotes the area of the triangle τ. Obviously, the elements of the
mass matrix are of order h2, where h is the diameter of the element τ.

Below we give the nodal basis function associated with the node (xi,yj)
(note we are using the notations (x,y) instead of (x1,x2)).

For the element τ1, which is a right triangle and the vertices are ordered
in the following way (P0, P1, P2) (see, Figure 3) we can easily compute the
element “stiffness” matrix A1τ :

A1τ =
1
2


 2 −1 −1−1 1 0
−1 0 1


 .

Assembling the local matrices will give the global matrix of the Ritz system.
For example, the equation for the internal point point P0 shown on Figure

6 MATH610: NUMERICAL METHODS FOR PDES – R. LAZAROV

finite element hφ0(x,y) h
∂φ0(x,y)

∂x
h
∂φ0(x,y)

∂y
τ1 h− (x−xi + y −yj) −1 −1
τ2 h− (y −yj) 0 −1
τ3 h + (x−xi) 1 0
τ4 h + (x−xi + y −yj) 1 1
τ5 h + (y −yj) 0 1
τ6 h− (x−xi) −1 0

Table 1. Analytic presentation of the nodal function at the
node 0 and its derivatives

3 will be

4U0 −U1 −U2 −U4 −U5 +
h2

12
(6U0 + U1 + U2 + U3 + U4 + U5 + U6)(3)

=
∫

Ω
f(x)φ0 dx.

Similarly, assembling the equation for the node P4 that is on the Neumann
boundary we get the equation:

2U4 −U0 −
1
2
U3 −

1
2
U8+

h2

24
(6U4 + 2U0 + U3 + 2U5 + U8)(4)

=
∫

Ω
f(x)φ4 dx.

References

[1] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, v. 19,
American Mathematical Society, 1991.

[2] Ch. Grossmann, H.-O. Ross, and M. Stynes, Numerical Treatment of Partial Differ-
ential Equations, Springer, Berlin, 2005.

[3] D. Kinkaid and W. Cheney, Numerical Analysis. Mathematics of Scientific Comput-
ing, Third Edition, Brooks/Cole, 2002.

[4] S. Larsen and V. Thomee, Partial Differential Equations with Numerical Methods,
Springer-Verlag, Texts in Applied Mathematics 45, 2003.

[5] M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Texts
in Applied Mathematics 13, Springer-Verlag, 1993.

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