Springer Series in

Computational
Mathematics

Elliptic
Differential
Equations

W. Hackbusch

Theory and
Numerical
Treatme...
Wolfgang Hackbusch

Elliptic

Differential
Equations

Theory and
Numerical
Treatment

Translated from the German

by Regin...
Wolfgang Hackbusch

MPI für Mathematik
in den Naturwissenschaften
lnselstr. 22-26
04103 Leipzig, Germany
e-mail: wh@mis.mp...
Foreword

This book has developed from lectures that the author gave for mathematics
students at the Ruhr-Universität Boch...
iv

Foreword

numerical treatment is the description of the discretisation procedures
(B), which give finite-dimensional e...
Table of Contents

Foreword
Thble of Contents

v

Notation
1

Partial Differential Equations and
Their Classification Into...
Vi

Table of Contents

4.6
4.7

62

Discretisations of Higher Order
The Discretisation of the Neumann
Boundary Value Probl...
vii

Table of Contents
6.2.4

H8(fl) for Real a 0

122

and Extension Theorems

123

Dual Spaces
6.3.1 Dual Space of a Nor...
viii

Thble of Contents
200

8.7.2 The Trefftz Method
8.7.3 Finite-Element Methods for Singular Solutions
8.7.4 Adaptive T...
Table of Contents
12

ix

Stokes Equations
12.1

12.2

275

275
SyBtems of Elliptic Differential Equations
278
Variational...
Notation

Formula Numbers.
Equations in Section X.Y are numbered (X.Y.1), (X.Y.2), etc. The equation (3.2.1) is referred t...
Notation

linear space of bounded operators from X to Y

L(X,Y)
L°°(S?)
L2(J2)
IN

n

n(x)

0
O(.)

P
qh

Rh,Rh

supp(f)
u...
1 Partial Differential Equations and
Their Classification Into Types

1.1 Examples
An ordinary differential equation descr...
2

1

Partial Differential Equations and Their Types

Let u be a solution. Introduce new coordinates = x + cy, = y and def...
1.1 Examples

3

(t4 —tii)/2.

w'= (u1 +t4,)/2,

From this one can determine and up to constants of integration. One
const...
4

1

Partial Differential Equations and Their Types

Example 1.1.9. (Cauchy-Riemann differential equations) If u and v sa...
1.2 ClassificatIon of Second-Order Equations into Types

5

Occasionally the parabolic type is defined only by ac — b2 = 0...
6

1 Partial Differential Equations and Their Types

which contains only the highest derivatives of L, is called the pri n...
of the Different

1.4 CharacteristIc

7

The Cauchy-Riemann system (1.14), which is dosely connected with the po-

tential...
8

1 Partial Differential Equations and Their Types

In Example 1.1.2 the hyperbolic differential equation (1.2) is augmen...
1.4 Characteristic Properties of the Different Types

9

(cf. Figure 2b) describes the temperature u(x, t) of a wire whose...
1 Partial Differential Equations and Their Types

10

00

u(z,y) =

a,,
.

sm(virx)sinh(viry).

1 and 0 y < 1, ti(z, y) is...
1.4 CharacterIstic Properties of the Different Types

ut+Lu=O
is a parabolic equation for u(x,t) =

11

(1.4.2)

In contra...
2 The Potential Equation

2.1 Posing the Problem
The potential equation from Example 1.1.3 reads

in.QCUt',

(2.1.la)

is ...
2.1 Posing the Problem

13

Definition 2.1.2. The function u is said to be harmonic in .17 if u belongs
and satisfies the ...
2 The Potential Equation

14

and u.,,, the radial derivative
This follows from the fact that, along with
also has to be b...
2.2 Singularity Function

15

The proof can be carried out directly. It is simplest to introduce polar
coordinates with y ...
____
16

2 The Potential Equation

PROOF. By
Kr(y) := (x E

x—

<r}

(2.2.7)

we denote the ball with centre y and radius ...
2.3 The Mean Value Property and Maximum Principle

u(y)

=j

(2.2.10)

dPi.

PROOF. (5b) implies

— u&P/OnJ

For a
weakeni...
18

2 The Potential Equation

Theorem 2.3.3. (Maximum-minimum principle) Let (1

be a domain
and let u E C°(Th be a noncon...
19

2.3 The Mean Value Property and Maximum Prmciple

Lemma 2.3.5. Harmonic functions have the mean value property.

PROOF...
2 The Potential Equation

20

The mean value property only assumes u E C°(Th, while harmonic functions belong to C2(.O) (1...
2.3 The Mean Value Property and Maximum PrInciple

21

According to Exercise 2.2.4 there exists a fundamental solution i'(...
2 The Potential Equation

22

Since
I

—

fro

fro

I
—

x—

f,.

it follows from Equation (8) that
Jo 5 max
Because

of t...
2.4 Continuous Dependence on the Boundary Data

23

2.4 Continuous Dependence on the Boundary Data
Definition 2.4.1. An ab...
2 The Potential Equation

24

The definition (1) of fl

1100

implies (3).

holds for a sequence of boundary values then T...
2.4 Continuous Dependence on the Boundary Data

Remark 2.4.5.

(a)

Let K be a set which is compact (i.e. complete and
E C...
26

2 The Potential Equation

(b) Let K C 1? be a compact set. Since 17,.. -+ 1' there exists a N(K) such
N(K)} converges
...
3 The Poisson Equation

3.1 Posing the Problem
The Poisson equation reads

au—f in!?

(3.1.la)

C°(Q). In the physical int...
3 The

28

Equation

3.2 Representation of the Solution by the Green
Function
Let the solution of (l.la,b) belong to C2(Th...
3.2 Representation of the Solution by the Green Function

u(x) =
—

x)

L

29

(3.2.2)

In the following we reverse the im...
3 The

30

Poisson Equation

The function I E Ck(1))

is

said to be k-fold Holder continuously

(with the exponent A), if...
32 Representation of the Solution by the Green Function

Theorem 3.2.11. Suppose the Green Junctiong(',x)
exists, and let ...
3 The Foieson Equation

32

Let

be defined by s(e, x) =

— xJ)

set

'

1

v(x)

(cf. (2.2.1)). For fixed r > 0
(3.2.8c)
...
____
3.2 Representation of the Solution by the Green Function

33

Finally we put forward two inequalities for the Green f...
3 The Poiseon Equation

34

The Green Function for the Ball

3.3

Theorem 3.3.1. The Green function for the bafl KR(y) is ...
3.4 The Neumann Boundary Value Problem

35

3.4 The Neumann Boundary Value Problem
were given on F. These
so-called Dirich...
36

Equation

3 The

Since u is only
Thus the term fru&y/Ondf in (2.1) becomes const•
determined up to a constant (cf. The...
3.5 The Integral Equation Method

&(x0) = 2u(xo),
for xo E F.
+ &(xo) =

37

(3.5.3a)

—

(3.5.3b)

In order that the ansa...
4 Difference Methods for the Poisson
Equation

4.1 Introduction: The One-Dimensional Case
Before developing difference met...
4.1 Introduction: The One-Dimensional Case

39

Lemma 4.1.1. Let {x—h,x-l-h)CTh. Then
= u'(x) i-hR with

if U E C2(Th)

8°...
40

Equation

4 Difference Methods for the

and tIh(l) with the aid of
If in (8a) one eliminates the components
Equation (...
4.2 The Five-Point Formula

41

In the Poisson equation

=f

—

in 11,

(4.2.2a)
(4.2.2b)

can each be replaced by the res...
4 Dlfferenee Methods for the Poisson Equation

42

Remark 4.2.1. fh is the restriction of f to the grid

For all points fa...
4.2 The

T

—I

—I

T

Lh=h2

4

4

43

—1

—1

—1

—1

.•.

...

..

—I

T

—1

—I

..•

Five-Point Formula

T

4

—1

—1...
44

4 DIfference Methods for the Poieson Equation

algorithm must take into account that Lh is sparse, i.e., it has substa...
45

4.3 M-matrices, MatriX Norms, Po6itive Definite Matrices

Definition 4.3.1. A is called an M-matrix if

forallaEI, Oap...
4 Difference Methods for the Poisson Equation

46

The important question as to whether A =

Lh is nonsingular can be

tre...
4.3 M-matrlces, Matrix Norms, Positive Definite Matrices

Since A is irreducible, for an arbitrary /3 E
applicable to
= /3...
48

4 Difference Methods for the Ponson Equation

In the following we split A into

A = D — B,

diag {a00:a €1),

D

(4.3....
4.3 M-rnatrices, Matrix Norms, Positive Definite Matrices
(4)

49

Alut if A 0.

Au lAut

The importance of inequality (7)...
50

4 Difference Methods for the Poisson Equation

Definition 4.3.12. Let V be a linear space (vector space) over the fiel...
4.3 M-matrices, Matrix Norms, Positive Definite Matrices

be the vector with the compo-

PROOF. As in the proof of Lemma 9...
52

4 Difference Methods for the Poisson Equation

For the proof in the exercise use the scalar product

(u,v) : CLUaVQ

(...
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Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathema...
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Wolfgang hackbusch elliptic_differential_equations_theory_and_numerical_treatment_springer_series_in_computational_mathematics,_18__2.003

  1. 1. Springer Series in Computational Mathematics Elliptic Differential Equations W. Hackbusch Theory and Numerical Treatment 4 Springer
  2. 2. Wolfgang Hackbusch Elliptic Differential Equations Theory and Numerical Treatment Translated from the German by Regine Fadiman and Patrick D.F. Ion With 40 Figures 4 Springer
  3. 3. Wolfgang Hackbusch MPI für Mathematik in den Naturwissenschaften lnselstr. 22-26 04103 Leipzig, Germany e-mail: wh@mis.mpg.de Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at httpd/dnb.ddb.de Second Printing 2003 Mathematics Subject Classification (2000): 35J20, 35J25, 35J30, 35J35, 35J50, 35J55, 65N06, 65N12, 65N15, 65N25, 65N30 ISSN 0179-3632 ISBN 3-540-54822-X Springer-Verlag Berlin Heidelberg New York o B.G. Teubner, Stuttgart 1987: W. Hackbusch, Throne und Numenik elliptischer Differentialgleichungen. Mit Genehmigung des Verlagea B.G. Teubner, Stuttgart, veranstaltete, allein autorisierte engliache Obersetzung der deutschen Originalausgabe. 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-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http-J/www.springer.de o Springer-Verlag Berlin Heidelberg 1992 Printed in Germany 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. Cover design: design&production. Heidelberg Printed on acid-free paper 46/3142oa-54321
  4. 4. Foreword This book has developed from lectures that the author gave for mathematics students at the Ruhr-Universität Bochum and the Christian-Albrechts-UniKiel. This edition is the result of the translation and correction of the Gennan edition entitled Theorie und NtLmerik Differentialgieichungen. The present work is restricted to the theory of partial differential equations of elliptic type, which otherwise tends to be given a treatment which is either too superficial or too extensive. The following sketch shows what the problems are for elliptic differential equations. B: Discretisatlon: Difference Methods, finite elements, etc. The theory of elliptic differential equations (A) is concerned with questions of existence, uniqueness, and properties of so&utions. The first problem of
  5. 5. iv Foreword numerical treatment is the description of the discretisation procedures (B), which give finite-dimensional equations for approximations to the solutions. The subsequent second part of the numerical treatment is numerical analysis (C) of the procedure in question. In particular it is necessary to find out if, and how fast, the approximation converges to the exact solution. The solution of the finite-dimensional equations (D, E) is in general no simpie problem, since from to 106 unknowns can oceur. The discussion of this third area of numerical problems is skipped (one may find it, e.g., in Hackbusch [5] and [9)). The descriptions of discretisation procedures and their analyses are closely connected with corresponding chapters of the theory of elliptic equations. In addition, it is not possible to undertake a well-founded numerical analysis without a basic knowledge of elliptic differential equations. Since the latter cannot, in general, be assumed of a reader, it seems to me necessary to present the numerical study along with the theory of elliptic equations. The book is conceived in the first place as an introduction to the treatment of elliptic boundary-value problems. It should, however, serve to lead the reader to further literature on special topics and and applications. It is intentional that certain topics, which are often handled rather summarily, (e.g., elgenvalue problems) are treated here in greater detail. The exposition is strictly limited to linear elliptic equations. Thus a discussion of the Navier-Stokes equations, which are important for fluid mechanics, i8 excluded; however, one can approach these matters via the Stokes equation, which is thoroughly treated as an example of an elliptic system. In order not to exceed the limits of this book, we have not considered further discretisation methods (collocation methods, volume-element methods, spectral methods) and Integral-equation methods (boundary-element methods). The Exercises that are presented, which may be considered as remarks without proofs, are an integral part of the exposition. If this book is used as the text for a course they can be used as student problems. But the reader too should test his understanding of the subject on the exercises. The author wishes to thank his collaborators G. Hofmann, G. Wittwn and J. Burmeister for the help in reading and correcting the manuscript of this book. He thanks Thubner Verlag for their cordial collaboration in producing the first German edition. Kiel, December 1985 W. Hackbusch This translation contains, in addition to the full text of the original edition, a short Section on the integral-equation method. The bibliography has also been expanded. The author wishes to thank the translators, R. Fadiman and P. D. F. Ion, for their pleasant collaboration, and Springer-Verlag for their friendly cooperation. Kiel, March 1992 W. Hackbusch
  6. 6. Table of Contents Foreword Thble of Contents v Notation 1 Partial Differential Equations and Their Classification Into Types 1.1 1.2 1.3 1.4 2 2.2 2.3 2.4 I . 4 . . . 6 . . . . 7 Posing the Problem Singularity Function The Mean Value Property and Maximum Principle Continuous Dependence on the Boundary Data . . 12 12 14 17 . . The Poisson Equation 3.1 3.2 3.3 3.4 3.5 4 1 Examples Classification of Second-Order Equations into Types Type Classification for Systems of First Order Characteristic Properties of the Different Types The Potential Equation 2.1 3 xi Posing the Problem Representation of the Solution by the Green Function The Green Function for the Ball The Neumann Boundary Value Problem The integral Equation Method 23 27 27 . 28 34 35 36 Difference Methods for the Poisson Equation 38 Introduction: The One-Dimensional Case The Five-Point Formula M-matrices, Matrix Norms, Positive Definite Matrices Properties of the Matrix Lh Convergence 38 4.1 4.2 4.3 4.4 4.5 40 44 53 59
  7. 7. Vi Table of Contents 4.6 4.7 62 Discretisations of Higher Order The Discretisation of the Neumann Boundary Value Problem 4.7.1 One-sided Difference for 9u/ôn 4.7.2 Symmetric Difference for Ou/On 4.7.3 Symmetric Difference for Ou/ôn 65 65 70 71 onanOffsetGrid Proof of the Stability Theorem 7 Discretisation in an Arbitrary Domain 4.8.1 Shortley-Weller Approximation 4.8.2 Interpolation at Points near the Boundary 72 4.7.4 4.8 5 78 78 . General Boundary Value Problems 5.1 5.2 5.3 Dirichlet Boundary Value Problems for Linear 85 Differential Equations 85 5.1.1 Posmg the Problem 86 5.1.2 Maximum Principle 5.1.3 Uniqueness of the Solution and Continuous 87 Dependence 5.1.4 Difference Methods for the General Differential 90 Equation of Second Order 95 5.1.5 Green's Function 95 General Boundary Conditions 95 5.2.1 Fbrmulating the Boundary Value Problem . . . Difference Methods for General Boundary 5.2.2 98 Conditions 103 Boundary Problems of Higher Order 103 5.3.1 The Biharmonic Differential Equation 5.3.2 General Linear Differential Equations of Order 2m 104 5.3.3 Discretisation of the Bihaimonic Differential . 105 Equation Ibols from Functional Analysis 6.1 6.2 83 85 . 6 . . . 110 Banach Spaces and Hubert Spaces 6.1.1 Normed Spaces 6.1.2 Operators 6.1.3 Banach Spaces 6.1.4 Hilbert Spaces Sobolev Spaces 110 6.2.1 L2(f1) 6.2.2 Hc(O) and 115 6.2.3 Fburier 110 111 112 114 115 116 and Hc(IRI)) 119
  8. 8. vii Table of Contents 6.2.4 H8(fl) for Real a 0 122 and Extension Theorems 123 Dual Spaces 6.3.1 Dual Space of a Normed Space 130 6.3.2 Adjoint Operators 6.3.3 Scales of Hilbert Spaces 132 6.4 Compact Operators 135 6.5 Bilinear Forms 137 Variational Ermulation 144 6.2.5 6.3 7 7.1 7.2 7.3 7.4 8 130 133 144 HIstorical Remarks Equations with Homogeneous Dirichiet Boundary Conditions Inhomogeneous Dirichiet Boundary Conditions Natural Boundary Conditions 145 150 152 The Method of Finite Elements 8.1 8.2 8.3 8.4 161 The Ritz-Galerkin Method Error Estimates FinIte Elements 8.3.1 Introduction: Linear Elements for 1? = (a,b) . 8.3.2 Linear Elements for a c JR2 8.3.3 Bilinear Elements for a C JR2 8.3.4 Quadratic Elements for (1 C It2 8.3.5 Elements for a C Ut3 8.3.6 Handling of Side Conditions Error Estimates for Finite Element Methods 8.4.1 H1-Estimates for Linear Elements 8.4.2 L2 and H Estimates for Linear Elements . Generalisations 8.5.1 Error Estimates for Other Elements 8.5.2 Finite Elements for Equations of Higher Order 8.5.2.1 Introduction: The One-Dimensional Btharmornc Equation 8.5.2.2 The Two-Dimensional Case 8,5.2.3 Estimating Errors Finite Elements for Non-Polygonal Regions Additional Remarks 8.7.1 Non-Conformal Elements . 8.5 8.6 8.7 161 167 171 . 171 174 178 180 182 182 185 185 . 190 193 193 . 194 194 195 196 196 199 199
  9. 9. viii Thble of Contents 200 8.7.2 The Trefftz Method 8.7.3 Finite-Element Methods for Singular Solutions 8.7.4 Adaptive Thangulation 8.7.5 Hierarchical Bases 201 . 201 202 202 8.7.6 Superconvergence 8.8 9 Properties of the 203 Matrix 208 RegularIty 9.1 Solutions of the Boundary Value Problem inH'(37),a>m 9.1.1 The Regularity Problem 9.1.2 Regularity Theorems for 9.1.3 Regularity Theorems for = 9.1.4 Regularity Theorems for General 1? C 9.1.5 Regularity for Convex Domains and Domains with Corners 9.1.6 Regularity in the Interior . Regularity Properties of Difference Equations 9.2.1 Discrete H'-Regularity 9.2.2 Consistency 9.2.3 Optimal Error Estimates 9.2.4 Hz-Regularity . 9.2 10 210 215 . 10.2 . Differential Equations with Discontinuous Coefficients 10.1.1 Fbnnulation 10.1.2 Discretisation A Singular Perturbation Problem 10.2.1 The Convection-Diffusion Equation 10.2.2 Stable Difference Schemes 10.2.3 Finite Elements 11.2 11.3 Formulation of Eigenvalue Problems . Finite Element Discretisation 11.2.1 Discretisation 11.2.2 Qualitative Convergence Results 11.2.3 Quantitative Convergence Results 11.2.4 Complementary Problems Discretisation by Difference Methods . 219 . 226 226 232 238 240 244 Elgenvalue Problems 11.1 . 223 226 SpecIal Differential Equations 10.1 11 . 208 208 244 244 246 247 247 249 251 253 253 . 254 254 256 . . . 260 264 . . 267
  10. 10. Table of Contents 12 ix Stokes Equations 12.1 12.2 275 275 SyBtems of Elliptic Differential Equations 278 Variational Formulation . 278 . 12.2.1 Weak Formulation of the Stokes Equations 279 12.2.2 Saddllepo4nt Problems 12.2.3 Existence and Uniqueness of the Solution of a 282 Saddlepoint Problem 285 12.2.4 Solvability and Regularity of the Stokes Problem 12.2.5 A Vo-elliptic Variational Formulation of the Stokes 289 Problem Mixed Finite-Element Method for the Stokes Problem . 290 12.3.1 Finite-Element Discretisation of a Saddlepoint 290 Problem 291 12.3.2 Stability Conditions 12.3.3 Stable Finite-Element Spaces for the Stokes Problem 293 12.3.3.1 Stability Criterion 293 12.3.3.2 Finite-Element Discretisations with the Bubble Function 294 12.3.3.3 Stable Discretisations with Linear Elements in Vh 296 12.3.3.4 Error Estimates 297 . 12.3 Bibliography 300 Index 307
  11. 11. Notation Formula Numbers. Equations in Section X.Y are numbered (X.Y.1), (X.Y.2), etc. The equation (3.2.1) is referred to within Section 3.2 8Irnply as (1). In other Sections of Chapter 3 it is called (2.1). Theorem Numbering. All Theorems, Definitions, Lemmata etc. are numbered together. In Section 3.2 Lemma 3.2.1 is referred to as Lemma 1. Special Symbols. The following quantities have fixed meanings: A, B,... B, B, C C'(D), Ck(D), C°°(D) Gr(a) d(u, VN) dl', diag{d1,d2,.. .} f h H"(i?), H'(S?), KR(y) I L I, matrices boundary differential operators (cf. (5.2.la,b), (5.3.6)) the complex numbers Holder- and Lipschitz-continuousiy differentiable functions (cf. Definition 3.2.8) k-fold and infinitely continuously differentiable functions infinitely differentiable functions with compact support.a (cf. (6.2.3)) distance of the function u from the subapace VN (ci. Theorem 8.2.1) surface differentials in surface integrals diagonal matrix with the diagonal eLements d1,d2,... a function; often the right-hand side of a differential equation Green's function (cf. Section 3.2) step size (cf. Sections 4.1 and 4.2) Sobolev spaces (cf. Sections 6.2.2 and 6.2.4) open ball about y with radius R (ci. (2.2.7), Section 6.1.1) identity or inclusion (cf. Sections 6.1.2, 6.1.3) a differential operator (cf. (1.2.6)) or the operator associated with a bilinear form (ci. (7.2.9')) the stiffnees matrix (ef. Section 8.1) matrix of a discrete system of equations (cf. (4.I.9a))
  12. 12. Notation linear space of bounded operators from X to Y L(X,Y) L°°(S?) L2(J2) IN n n(x) 0 O(.) P qh Rh,Rh supp(f) u = u(x) = u(xj, Uh VN,Vh x,(x,y),(x,y, z) x=(xl,...,xn) z I, p(A) 1? V 8; 8/On )xXx' (., .) (., .)O, (, (•, (•, )a II112 • j I1•1100 I I I - 1k, I lo, I (cf. Section 6.1.2) space of essentially bounded functions (cf. 6.1.3) space of square-integrable functions (cf. Section 6.2.1) the natural numbers {1,2,3, . . normals (ci. (2.2.3a)) the zero matrix constlg(x)I Landau symbol: 1(x) = O(g(x)) if cf. (8.1.6) a grid function, nght-hand side of the discrete equation (4.1 .9a) the real numbers, the positive real numbers restrictions (cf. (4.5.2) and (4.5.5b)) the singularity (unction (cf. Section 2.2) the support of the function f (cf. Lemma 6.2.2) a function, e.g., a solution of a differential equation a grid function (discrete solution; cf Section 4) finite-element spaces (ci. (8.1.3) and Section 8.4.1) independent variables a vector of independent variables the integers the boundary of 1? the boundary points of a grid (cf. (4.2.lb), (4.8.4)) fundamental solution function spectral radius of the matrix A or a domain (cf. DefinitIon 2.1.1) an open set In a grid (cf. (4.1.6a), (4.2.la), (4.8.2)) the Laplace operator (cf. (2.1.la)) the five-point difference operator (ci. (4.2.3)) gradient (cf. (2.2.3a)) differences (cf. (4.1.2a-c), (4.2.3)) differences in the normal direction (cf. (4.7.4)) normal derivative (cf. (2.2.3a)) acalar product (cf. (2.2.3c), (4.3.14a)) duality form (ci. Section 6.1.3) acalar product (cf. Section 6.1.4) scalar product and norms on L2(Q) scalar products and norms on the Eudidean norm (cf. (2.2.2)) the Eucidean norm and the spectral norm (cf. Section 4.3) norms equivalent to J , (cf. (6.2.15), (6.2.16b)) maximum norm (ci. (4.3.3)), row sum norm (4.3.11), or supremum norm (2.4.1), ci. also Section 6.1.1 the vector (1, 1,...) (ci. Section 4.3) J
  13. 13. 1 Partial Differential Equations and Their Classification Into Types 1.1 Examples An ordinary differential equation describes a function which depends on only one variable. Unfortunately, for many problems it is not possible to restrict attention to a single variable. Almost all physical quantities depend on the spatial variables z, y , and z and on time t. The time dependence might be omitted for stationary processes, and one might perhaps save one spatial dimension by special geometric assumptions, but even then there would still remain at least two independent variables. Equations that contain the first partial derivatives where (1 i or even higher partial derivatives etc., are called partial differential equations. Unlike ordinary differential equations, partial differential equations cannot be analysed all together. Rather, one distinguishes between three types of equations which have different properties and also require different numerical methods. Before the characteristics for the types are defined, let us introduce some examples of partial differential equations. All of the following examples will contain only two independent variables x,y. The first two examples are partial differential equations of first order, since only first partial derivatives occur. Example 1.1.1. Find a solution u(x, y) of (1.1.1) It is obvious that u(x, y) must be independent of y, i.e., the solution has the form u(x, y) = Thus u(x, y) = w(x) for some arbitrary is a solution of (1). Equation (1) is a special case of Example 1.1.2. Find a solution u(x, y) of (cconstant). (1.1.2)
  14. 14. 2 1 Partial Differential Equations and Their Types Let u be a solution. Introduce new coordinates = x + cy, = y and define I?)) with the aid of =— = Since v(e, := ii), (chain rule) and X,3 —c, y,7 = 1, it follows from (2) that V,3 = + = 0. This equation is analogous to (1), and Example 1 shows that v(e, = co(e). If one now replaces ,g by y one obtains u(x,y) = (1.1.3) Conversely, through (3) one obviously obtains a solution of Equation (2) as long as is continuously differentiable. In order to determine uniquely the solution of an ordinary differential f(u) = 0 one needs an initial value u(x0) u0. The partial differential equation (2) can be augmented by the initial-value function equation t&' — for x€Ut (1.1.4) with aconstant. The comparison of Equations (3) and on the line (4) shows that = uo(x). Thus is determined by = The unique solution of the initial value problem (2) and (4) reads u(x, y) = tio(x — c(yo — y)). (1.1.5) The following three examples involve differential equations of second order. Example 1.1.3. (Potential equation or Laplace equation) Let 0 be an open subset of Find a solution of infi. (1.1.6) If one identifies (z, y) E It2 with the complex number z = x + iy C, the solutions can be given immediately. The real and imaginary parts of any function f(z) holomorphic in flare solutions of Equation (6). Three examples are Rez0 = 1, Rez2 = x2—y2 and Relog(z—zo) if $1. To determine the solution uniquely one needs the boundary values for all (x,y) on theboundaryf=Ofloffl. Example 1.1.4. (Wave equation) All solutions of (1.1.7) are given by u(x,y) = w(x + y) + — y) (1.1.8) where and are arbitrary twice continuously differentiable functions. Suitable initial values are, for example, u(x,O) = uO(x), where = p+ u1(z), (x E It) (1.1.9) and u1 are given functions. If one inserts (8) into (9), one finds tq = — , where is the derivative of and infers that
  15. 15. 1.1 Examples 3 (t4 —tii)/2. w'= (u1 +t4,)/2, From this one can determine and up to constants of integration. One constant can be chosen arbitrarily, for example, by s°(O) = 0, and the other is = determined by ti(O, 0) = ço(0) + Exercise 1.1.5. Prove that every solution of the wave equation (7) has the Example 1.1.6. (Heat equation) Find the solution of (1.1.10) (physical interpretation: u = temperature, y = time). The separation of variables u(z, y) = v(x)w(y) gives that for every c E lit (1.1 .lla) sin(cx) . exp(—c2y). u(x, y) Another solution of (10) for y > 0 is u(x y) = L (l.l.Ilb) exp((z — is an arbitrary continuous and bounded function. The initial condition matching Equation (10), in contrast to (9). contains only one function: where u(x,0) = uO(x). (1.1.12) The solution (lib), which initially is defined only for y > 0, can be extended continuously to y =0 and there satisfies the initial value requirement (12). ExercIse 1.1.7. Let uo be bounded in lit and continuous at x. Then prove that the right side of Equation (lib) converges to uo(z) for y —'0. Hint: First show that tz(x, y) = 110(z) + — 110(z)) exp( — and then decompose the integral into subintegrals over [z — 6,z +5) and (—oo,z—S)U(z+6,oo). As equations, equations of higher order can be by systems of first-order equations. In the following we give some with ordinary differential described examples. Example 1.1.8. Let the pair (u, v) be the solution of the system (1.1.13) If u and v tions are + twice =0, differentiable, + the differentiation of (13) yields 0, which together imply Thus u is a solution of the wave equation (7). The same that the equa- - =0. be shown for v.
  16. 16. 4 1 Partial Differential Equations and Their Types Example 1.1.9. (Cauchy-Riemann differential equations) If u and v satisfy the system v2—u5=O in $?cIt2, (1.1.14) then the same consideration as in Example 8 yIelds that both u and v satisfy the potential equation (6). Example 1.1.10. If u and v satisfy the system v1+u=0, (1.1.15) then v solves the heat equation (10). A second-order system of interest in fluid mechanics can be found in Example 1.1.11. (Stokes equations) In the system (1.1.16a) + — = 0, (1.1.16b) =0 (1.1.16c) u and v denote the flow velocities in x- and y-directions, while w denotes the pressure. 1.2 Classification of Second-Order Equations into Types The general linear differential equation of second order in two variables reads +d(x,y)u +e(x,y)uy 1-f(x,y)u-f-g(z,y) = ( 1. Q 2. i ) DefinitIon 1.2.1. (a) Equation (I) is said to be elliptic at (x,y) if a(x,y)c(z,y) — b(x,y)2 > 0. (1.2.2a) (b) Equation (1) is said to be hyperbolic at (x,y), if a(x,y)c(z, y) — b(x,y)2 <0. (1.2.2b) (c) Equation (1) is said to be parabolic at (x,y) if ac—b2=0 and rank[ b =2 at(z,y). (1.2.2c) (d) Equation (1) is said to be elliptic (hyperbolic, parabolic) in ii C 1R2 if it is elliptic (hyperbolic1 parabolic) at all (x, y) E fl
  17. 17. 1.2 ClassificatIon of Second-Order Equations into Types 5 Occasionally the parabolic type is defined only by ac — b2 = 0. But one y) = 0 parabolic, nor y) + would not want to call the equation indeed the purely algebraic equation u(z, y) =0. Example 1.2.2. The potential equation (1.6) is elliptic, the wave equation (1.7) is of hyperbolic type, while the heat equation (1.10) is parabolic. The definition of types can easily be generalised to the case where more . The general linear dif, than two independent variables occur, say reads ferential equation of second order in n variables x = (x1,.. . , . + a(x)u = 1(x). + (1.2.3) = holds for twice continuously differentiable functions, one can assume that, without loss of generality, Since aj3(x) = a,1(x). (1.2.4a) define a symmetric n x n matrix Thus, the coefficients A(x) = (1.2.4b) which therefore has only real eigenvalues. Definition 1.2.3. (a) Equation (3) is said to be elliptic at x if all n elgenvalues of the matrix A(x) have the same sign (±1) (i.e. if A(x) is positive or negative definite). (b) EquatIon (3) is said to be hyperbolic at x if n — 1 eigenvalues of A(x) have the same sign (±1) and one eigenvalue has the opposite sign. (c) Equation (3) is said to be parabolic at x if one eigenvalue vanishes, the remaining n —2 eigenvaluen have the same sign, and rank(A(x), a(x)) = n where a(x) = (aj(x), .. . , (d) Equation (3) is said to be elliptic in Si c if it is elliptic at all x 0. Definition 3 makes it clear that the three types mentioned by no means cover all cases. An unclassified equation occurs, for example, if A(x) has two positive and two negative eigenvalues. In place of (3) one can also write Lu=f, (1.2.5) where (1.2.6) is a linear differential operator of second order. The operator
  18. 18. 6 1 Partial Differential Equations and Their Types which contains only the highest derivatives of L, is called the pri nd psi part of L. Remark 1.2.4. The ellipticity or hyperbolicity of Eq. (3) depends only on the principal part of the differential operator. ExercIse 1.2.5. (Invariance of the type under coordinate transformations) Let Eq. (3) be defined for x a. The transformation i' fl C IR" SI C 1W' is assumed to have a nonsmgular Jacobian matrix S = E C1(Q) at x. Prove that Eq. (3) does not change its type at x if it is written in the new coordinates = Hint: the matrix A = (ad) becomes SAST after the transformation. Use Remark 4 and Sylvester's inertia theorem (cf. Gantmacher 11, Chapter 1.3 Type Classification for Systems of First Order The examples 1.1.8-10 are special cases of the general linear system of first order in two variables. u.,(x,y)— A(x,v)uv(x,v)÷B(x,y)u(x,y) = f(x,y). (1.3.1) Here, u=(uj,...,um)Tisavectorfunction,andA,Baremxrnmatrjces.In contrast to Section 1.2, A need not be symmetric and can have complex elgenvalues. If the elgenvalues A1,... , are real, and if there exists a decomposition A = with D = diag{A1,. .. ,A,,,}, A is called real-diagonalisable. DefinitIon 1.3.1. (a) System (1) is said to be hyperbolic at (x,y) if A(:,y) is real-diagooalisable. (b) System (1) is said to be elliptic at (x,y) if all the elgenvalues of A(x, y) are not real. If A is real or possesses m distinct real eigenvalues the system is hyperbolic since these conditions are sufficient for real diagonalisability. A single real equation, in particular, is always hyperbolic. Examples 1.1.1 and 1.1.2 are hyperbolic according to the preceding remark. System (1.13) from Example 1.1.8 has the form (1) with A={01 It is hyperbolic since A is real-diagonalisable: A— [1 ii [—1 0111 1 1
  19. 19. of the Different 1.4 CharacteristIc 7 The Cauchy-Riemann system (1.14), which is dosely connected with the po- tential equation (1.6), is elliptic since it has the form (1) with A={? the eigenvalues ±i. The system (1.15) corresponding to the (parabolic) heat equation can be described as system (1) with and A has The eigenvalues (A1 = A2 =0) may be real but A is not diagonalisable. Hence, system (1.15) is neither hyperbolic nor elliptic. A more generaL system than (1) is + + 13u = f. (1.3.2) If A1 is invertible then multiplication by Aj' gives the form (1) with A —A' A2. Otherwise one has to investigate the generalised eigenvalue problem det(XA1 + A2) = 0. However, system (2) with singular A1 cannot be elliptic, as can be seen from the following (cf. (4) with = 1, = 0). A generalisation of (2) to n independent variables is exhibited in the system (1.3.3) with m x m matrices . and B = B(x). As a special case of a later definition (cf. Sect. 12.1) we obtain the Definition 1.3.2. The system (3) is said to be elliptic at x if for any vector o . the following holds: 0. 1.4 (1.3.4) Characteristic Properties of the Different Types The distinguishing of different types of partial differential equations would be pointless if each type did not have fundamentally different properties. When discussing the examples in Sect. 1.1 we already mentioned that the solution is uniquely determined if initial values and boundary values are prescribed.
  20. 20. 8 1 Partial Differential Equations and Their Types In Example 1.1.2 the hyperbolic differential equation (1.2) is augmented by the specification (1.4) of u on the line y = oonst (see Fig. la). In the case of the hyperbolic wave equation (1.7), it.,, must also be prescribed (cf. (1.9)) since the equation is of second order. (a) u,u,,, (b) u,u,,, FIgure 1.4.1. (a) Initial conditions and (b) initial boundary conditions for hyperbolic problems if it is also sufficient to give the values it and it,1 on a finite interval [xj, u is additionally prescribed on the lateral boundaries of the domain I? of Fig. lb. This prescription of initial boundary values occurs, for example, in the following physical problem. A vibrating string is described by the lateral deflection u(x,t) at the point x (xl,z21 at time t. The function it satisfies the wave equation (1.7) with the coordinate y corresponding to time t. At the are initial instant in time, t = to, the deflection u(x,O) and velocity given for x1 <x <x2. Under the assumption that the string is firmly damped at the boundary points xl and x2, one obtains the additional boundary data u(xi,t) =t4x2,t) =0 for alit. For parabolic equations of second order one can also formulate initial value and initial boundary value problems (cf. Fig. 2). However, as initial value only the function = tlO(x) may be prescribed. An additional is not possible, since specification of is already determined by the differential equation (1.10) and by it0. (a) Initial conditions and (b) initial boundary conditions for parabolic problems The heat equation (1.10) with the initial and boundary values t4x,to)=uo(x) u(xj,t) in [xi,X21, ti(x21t) = fort >
  21. 21. 1.4 Characteristic Properties of the Different Types 9 (cf. Figure 2b) describes the temperature u(x, t) of a wire whose ends at x = and x = x2 have the temperatures distribution at time to is given by uO(x). and The initial temperature Aside from the different number of initial data functions in Figure 1 and Figure 2, there also is the following difference between hyperbolic and parabolic equations. Remark 1.4.1. The shaded area 11 in Figure 1 corresponds to t > to, and For hyperbolic equations one can solve in the same way in Figure 2 to y > initial-value and initial-boundary-value problems in the domain t to, whilst parabolic problems in t <to generally do not have a solution. If one changes the parabolic equation Ut — = 0 to + orientation is reversed: solutions exist in general only for t to. = 0, the For the solution of an elliptic equation, boundary values are prescribed (cf. Example 1.1.3, Figure 3). A specification such as that in Figure 2b would not uniquely determine the solution of an elliptic problem, while the solution of a parabolic problem would be overdetermined by the boundary values of Figure 3a.. U (a) U. u (b) FIgure 1.4.3. Boundary value conditions for an elliptic problem An elliptic problem with specifications such as in Figure lb in general has no solution. Let us, for example, impose the conditions u(x, 0) = u(0, y) u(1,y) = 0 and = on the solution of the potential equation (1.6), where u1 is not infinitely often differentiable. If a continuous solution u existed in 17 = [0,1j x [0,11 one could develop u(z,1) into a sine series and the following exercise shows that uj would have to be often differentiable, in contradiction to the aseumption. Exercise 1.4.2. Let E C°[0, 1) have the Fourier expansion 00 = Show that: (a) the solution of the potential equation (1.6) in the square (1 = (0,1) x (0,1) with boundary values u(O,y) = u(x,0) = u(1,y) = 0 and u(z, 1) = is given by
  22. 22. 1 Partial Differential Equations and Their Types 10 00 u(z,y) = a,, . sm(virx)sinh(viry). 1 and 0 y < 1, ti(z, y) is differentiable infinitely often. Hint: (b) For 0 f(x) = E13,,sin(ziwz) E C°°[0, 1), if lim/3,t) = 0 for all k E IN. Conversely it does not make sense to put boundary value constraints as in Fig. 3a on a hyperbolic problem. Consider as an example the wave equation (1.7) in = [0, 1) x [0, 1/711 with the boundary values u(x,O) = IN. The solution ii(0, y) = u(1, y) = 0 and u(x, 1/jr) = sin(virx) for v reads u(x, y) = sin(v7rx) sin(viry)/ sin ii. Although the boundary values, for all p IN are bounded by 1, the solution in I? may become arbitrarily large since sup{1/sinv: v IN} = 00. Such a boundary value problem is called "not well posed" (cf. Definition 2.4.1). Exercise 1.4.3. Prove that the set {sinv:v IN) is dense in [—1, 1]. Another distinguishing characteristic is the regularity (smoothness) of the solution. Let u be the solution of the potential equation (1.6) in Q C 1R2. As stated in Example 1.1.3, ti is the real part of a function holomorphic in Si. Since holomorphic functions are infinitely differentiable, this property also holds for ti. In the case of the parabolic heat equation (1.10) with boundary values u(z,0) = no the solution z* is given by (1.llb). For y >0, ti is infinitely differentiable. The smoothness of is of no concern here, nor is the smoothness of the boundary values in the case of the potential equation. One finds a completely different result for the hyperbolic wave equation (1.7). The solution reads u(z,y) = and result — y), where directly from the initial data (1.9). Check that u is k-times differentiable if u0 is k-times and u1 is (k — 1)-times differentiable. As already mentioned in this section, in the hyperbolic and parabolic equations (1.1), (1.2), (1.7) and (1.10) the variable y often plays the role of time. Therefore one calls processes described by hyperbolic and parabolic equations nonstationary. Elliptic equations which only contain space coordinates as variables are called stationary. More clearly than in Definitions 1.2.lb,c the Definitions 1.2.3b,c distinguish the role of a single variable (time) corresponding to the elgenvalue A = 0 in parabolic equations, and to the elgenvalue with opposite sign in hyperbolic equations. The connection between the different types becomes more comprehensible if one relates the elliptic equations in the variables to the parabolic and hyperbolic equations in the variables x1, ,x,,,t. Remark 1.4.4. Let L be a differential operator (2.6) in the variables x = (x1, . . , and let it be of elliptic type. Let L be scaled such that the matrix A(x) in (2.4b) has only negative eigenvalues. Then .
  23. 23. 1.4 CharacterIstic Properties of the Different Types ut+Lu=O is a parabolic equation for u(x,t) = 11 (1.4.2) In contrast utt+Lu=O (1.4.3) is of hyperbolic type. Conversely, the nonstationary problems (2) or (3) lead to the elliptic equation Lu = 0 if one seeks solutions of (2) or (3) that are independent of time t. One also obtains elliptic equations if one looks for solutions of (2) or (3) with the aid of a separation of variables u(x, t) ço(t)v(x). The are u(x,t) = in case (2), incase(3), (1.4.4) where v(x) is the solution of the elliptic eigen value problem Lv=Av. (1.4.5)
  24. 24. 2 The Potential Equation 2.1 Posing the Problem The potential equation from Example 1.1.3 reads in.QCUt', (2.1.la) is the Laplace operator. In physics, + 02/Ox? + Equation (la) describes the potentials; for example the electric potential when a contains no electric charges, the magnetic potential for vanishing current where density, the velocity potential, etc. Equation (is) is also called Laplace's equation since it was described by P. S. Laplace in his five-volume work "Mécanique Céleste" (1799-1825). However, It was L. Euler who first mentioned the potential equation in 1752. The connection between the potential equation for n = 2 and function theory has already been pointed out in Example 1.1.3. Not only is the Laplace operator an example of an elliptic differential operator, but it actually is the prototype (the so-called normal form). By using a transformation of variables, any elliptic differential operator of second order can be transformed so that its principal part becomes the Laplace operator (cf. Hellwig Li, in the following a will alwayB be a domain, Definition 2.1.1. The region a C lit" is called a domain if a is open and connected.1 The existence of a second derivative of u is required only in 0, not on the boundary r=aa of 0. For a prescription of boundary values u=cp onE' (2.l.lb) to be meaningful, one has to aseume continuity of u on considerations lead to 1 1? Is called connected If for any x,y E 0 that connects x and y, thai is, a -y: $ 'y(O) = x,"y(l) = y. = 0 U P. These (1 there is a continuous curve within '7(s) E continuously with (0,11 0
  25. 25. 2.1 Posing the Problem 13 Definition 2.1.2. The function u is said to be harmonic in .17 if u belongs and satisfies the potential equation (la). to C2($?) fl Here C°(D) denotes the set of continuous fk-fold continuously differentiable, infinitely often differentiablej functions on D. In general one should not expect that the solution of (la,b) lies in as we show in the following example. Example 2.1.3. Let $7 = (0, 1) x (0,1) (cf. Figure 1.4.3a). Let the boundary values be given by (x, y) E I'. A solution of the boundary value problem exists but does not belong to C2(.Q). PROOF. The existence of a solution u will be discussed in Theorem 7.3.7. If u E C2(Th, then it follows that = = 2 for x 10,1); in particular 0) = 2. From the analogous result 0) = 0) = 0 one may conclude Llu(0, 0) = 2 in contradiction to = 0 in .17. U In the case under discussion one can also show u statement is generally false, is shown by Example 2.1.4. In the domain .17 = (—i, x (—i, Figure 1) introduce the polar coordinates y=rsinço. That this [0, x [0, (cf. (2.1.2) ft 1'l ft ft FIgure 2.1.1. An 1,-shaped region The function u(r, = is the solution of the potential equation (la) and has smooth boundary values on F (in particular u = 0
  26. 26. 2 The Potential Equation 14 and u.,,, the radial derivative This follows from the fact that, along with also has to be bounded, However, u,. = O(r1/3) holds 0. In order to check that 41u =0, use the following. = for r -. Show that (a) In terms of the polar coordinates (2) in lit2, the Laplace operator takes the form ExercIse 2.1.5. 182 18 82 (2.1.3) (b) In terms of the three-diniensional polar coordinates y=rsinçosint,b, the Laplace operator is given by 82 28 1 1 8 82 82 (2.1.4) Note. In the n-dimensional case the transformation to polar coordinates leads to n—18 82 1 (2.1.5) where the Beltrami operator B contains only derivatives with respect to the angle variables. 2.2 Singularity Function The singularity function is defined by forn=2, s(x,y) := — I.. for x,y fOrn> 2 (n—2)w,, (2.2.la) lit", where = /F(n/2), = 2w, = 4w, (2.2.Ib) with I' the Gamma function, is the surface of the n-dimensional unit sphere. The Euclidean norm of x in IR" is denoted by 1/2 (xl by = (2.2.2) 2.2.1. RrJlxed y E Ut" the potential equation in 1R"{y} is 8olved s(x,y).
  27. 27. 2.2 Singularity Function 15 The proof can be carried out directly. It is simplest to introduce polar coordinates with y as origin and to use (1.5), since s(x,y) depends only on r=lx—yI. For the next theorem we need to introduce the normal derivative 8/On. denote the outer F, i.e. n is a unit vector perpendicular to the tangential hyperplane at x and points outwards. The normal derivative of u at x E F is defined as be a domain with smooth boundary 1'. Let n(x) E Let normal direction at x = (n(x), Vu(x)), (2.2.3a) where (2.2.3c) is the gradient of u and (x,y) = (2.2.3c) is the scalar product in Ut". In the case of the sphere KR(y) (cf. (7)) the normal direction is radial, and Ou/On becomes Ou/Or with respect to r = — y11 if one uses polar coordinates with the origin at y. It follows from Os(x, y)/Or = —Ix — that Os(x,y)/On = for x E (2.2.4) The first Green formula reads (cf. Green [1J) L u dx = — V E j (Vu, Vv) + j if the domain 1? satisfies suitable condi- tions. Here frn... dl' denotes the surface integral. Domains for which Equation (5a) holds are called normal domains. To see sufficient conditions for this refer to Kellogg (1, Chapter LVI and Hellwig 11, 1-1.21. Functionsu,v E C2(Thinanormaldomain.Qsatisfythe second Green formula / u4v = / viju + / — dl'. (2.2.5b) Theorem 2.2.2. Let (2 be a normal domain, and Let u E there. Then u(y) = for all y (2. Here 0/On, and dl' refer to the variable x. be harmonic dl', (2.2.6)
  28. 28. ____ 16 2 The Potential Equation PROOF. By Kr(y) := (x E x— <r} (2.2.7) we denote the ball with centre y and radius r. Since the singularity function s(., y) is not differentiable on x = y, Green's formula (5b) is not directly applicable. Let := with e so small that from = Lts = 0 C £7. Since £7 is again a normal domain, it follows (ci. Lemma 1) and (5b) with v a that J = 0. — (2.2.8a) the normal directions However, at x E = 8flU We have and of 8K,, (y) differ in their signs. The same holds for the normal of ... = derivatives, so that the integral in (8a) can be decomposed into The assertion of the theorem would be proved if we — —. —u(y) for e 0. The normal derivative 8u/On could show that and fSK(y) s(x, y) di converges like 0(eI log or is bounded on Thus, 0(e) towards zero, as can be seen from (1) and f8K,(y) dl" = we obtain From fSK 8 dl = --.0 (e —-.0). (2.2.8b) and (4) one infers 8 1 J8K,(y) y) di', = —u(y). (2.2.8c) — —.0 as e —e 0. V74X The continuity of u in y yields J (u(x) max — xEOK1(y) (2.2.8d) Equations 8b,c,d show that fSK(y) dl' —' —u(y) (e —. 0), so — that (8a) proves the theorem. Any function of the form lr(x, y) = 8(X, y) + y) (2.2.9) is called a fundamental solution, of the potential equation, in .17 if for fixed y E .1'? the function is harmonic in .0 and belongs to C2(Th). Corollary 2.2.3. Under the conditions of Theorem 2 for solution in.0 the following holds: fundamental
  29. 29. 2.3 The Mean Value Property and Maximum Principle u(y) =j (2.2.10) dPi. PROOF. (5b) implies — u&P/OnJ For a weakening ofthecondition C'(FJ) fl C2(.Q) refer to Heliwig (1, t—1.4J. 17 df =0. U = -y—sE C2(Th) for E Exercise 2.2.4. Let 1? .= KR(y). Define — [Ix = for — n> 2 (2.2.lla) for E Q, =y+ — — y) and show: (a) -y is a fundamental solution in fl, (b) y(x, = x), (c) on the surface 1' = OKR(y) the following holds: with = = — Ix — y12)Ix E I'). (2.2.llb) 2.3 The Mean Value Property and Maximum Principle DefinitIon 2.3.1. A function u has the mean value property in S? if C°(fl) and if for all x E Q and all R >0 with KR(X) C a the following equation holds UE u(x) 1 J = (2.3.1) 8KR(x) dl.' = the right side in (1) is the mean value of u taken over the surface of the sphere. An equivalent diaracterisation results if Since one averagea over the sphere KR(x). Exercise 2.3.2. u C0(IJ) has the second mean value property in 1? if u(x) = for all x a, R> 0 with KR(x) C a. Show that this mean value property is equivalent to the mean value property (1). Hint: IKR(x) = fR fSK(x) dr. Functions with the mean value property satisfy a maximum principle, as is known from the function theory for holomorphic functions:
  30. 30. 18 2 The Potential Equation Theorem 2.3.3. (Maximum-minimum principle) Let (1 be a domain and let u E C°(Th be a nonconstant function which has the mean value property. Then u takes on neither a maximum nor a minimum in a. PROOF. (a) It suffices to investigate the case of a maximum since a minimum of u is a maximum of —u, and —u also has the mean value property. (b) For an indirect proof we assume that there exists a maximum in y E a: u(y)=Mu(x) fora.llxEa. In (c) we will show u(y') = M for arbitrary y' E I?, i.e. u M in contrast to the assumption u const. (c) Proof of u(y') = M. Let y' E (1. Since S? is connected, there exists a path connecting y and y' running through a, i.e. there exists a continuous (2 with y, 10,11 = y'. We set I := {s M for all 0 t s}. E 10,11: I contains at least 0, and is closed since u and ço are continuous. Thus there In (4) exists s' = max{s I}, and the definition of I shows that I = [0, it is proved that s' =lso that y= E land hence u(y') =M follows. (d) Proof of? = 1. The opposite assumption 8' <lean be made and shown to be contradicted by proving that ii(x) = M in a neighbourhood of x' := Since x' (2, there exists R> 0 with C £1. Evidently, it follows that u = M in KR(x)' if it is shown that u = M on ôKr(X)' for all 0 < r < R. (e) Proof of u = Mon OKr(x'). Equation (1) in x' reads M = u(x') = I JSKr(X') In general we have u(C) M . If one had u(C') <M fore' OKr(X') and thus also u < M in a neighbourhood of one would have on the right side a mean value smaller than M. Therefore, u = M on ÔK4x') has been proved. • Simple deductions from Theorem 3 are contained in Corollary 2.3.4. Let (2 be bounded. (a) A function with the mean value property takes its maximum and its minimum on 8(2. (b) If two fsinctioni with the mean value property coincide on the boundary 8, they are identicaL PROOF. (a) The extrema are assumed on the compact set = QU Ofl. According to Theorem 3, the extremum cannot be in flif u is not constant on a connected component of fl. But in this case the assertion is also obvious. (b) If u and v with u = v satisfy the mean value property on 8$? then the latter is also satisfied for w := u v. Since w = 0 on 0(2, part (a) indicates -
  31. 31. 19 2.3 The Mean Value Property and Maximum Prmciple Lemma 2.3.5. Harmonic functions have the mean value property. PROOF. Let u be harmonic in Qand y KR(y) C (2. We apply the representation (2.6) for (2 = KR(y). The value s(x,y) is coostant on 8KR(y): it be denoted by a(R). Because of (2.4), Equation (2.6) becomes f Ott u(y) = cr(R) j dl' + 1 1 udf'. J8KR(y) The equation agrees with (1) if the first integral vanishes. The latter follows from Lemma 2.3.6. Let u E C2(Th) be harmonic in a normal domain 0. Then we have (2.3.2) PROOF. In Green's formula (2.5a) substitute 1 and u for uand v respectively. S Lemma 5, Theorem 3, and Corollary 4 together imply Theorems 7 and 8: Theorem 2.3.7. (The maximum-minimum principle for harmonic functions) Let u be harmonic in the domain $1 and nonconstant. There exists no maximum and no minimum in 11. Theorem 2.3.8. (Uniqueness). Let $2 be bounded. A function harmonic in I? assumes its maximum and its minimum on 8$? and is uniquely determined by its values on 811. The representation (1) of u(y) by the values on OKR(y) is a special case of the following formula which will be proved at the end of this section and which provides Equation (1) for x = y. Theorem 2.3.9. (Poi8son's integral formula) Assume we have cc E C°(OKR(y)) and n 2. The solution of the boundary value problem 4u = 0 in KR(y), u= cc on OKR(y) (2.3.3) is given by the function u(x) = R2_Ix_. Yl 2 f 8KR(y) IX — which belongs to C°°(KR(y)) Ii C°(KR(y)). for x KR(y), (2.3.4)
  32. 32. 2 The Potential Equation 20 The mean value property only assumes u E C°(Th, while harmonic functions belong to C2(.O) (1C°(fl). This makes the following assertion surprising: Theorem 2.3.10. A function is harmonic infl if and only if it has the mean value property there. PROOF. Because of Lemma 5 it remains to be shown that a function v with the mean value property is harmonic. Let x E KR(X) C (1 be given arbitrarily. According to Theorem 9 there exists a function u harmonic in KR(X) with inKn(x), u=v on.ÔKR(x). According to Lemma 5, u has the mean value property, as does v, and Corol- lary 4b proves that u = i.e. v is harmonic in KR(X). Since v in U KR(X) C 11 is arbitrary, v is harmonic in (1. An important application of Theorem 10 is Theorem 2.3.11. (Harnack) Let monic in 1? and converging uniformly in be a sequence of functions harUk is harmonic = Then u mi?. The limit process, PROOF. The limit function is continuous: u yields Equation (1) for u; i.e. applied to Uk(X) = u has the mean value property. According to Theorem 10, u is also harmonic N intl. Theorems 3 and 7 on the maximum-minimum principle pertain to global extrems. The proof of Theorem 3 does not yet exclude local extrema in the interior. It merely shows that u is then always constant in a circle KR(y) C 1?. As is known from function theory one can derive from this u(x) = M in if u is analytic, 1. e., there is a convergent power series expansion in a neighbourhood of any X E I?. Indeed, the following theorem holds whose proof can be found, for example, in Hellwig (1,11111.5]: Theorem 2.3.12. A function harmonic in tl is analytic there. The proof of the Poisson formula still needs to be carried out. (a) First we = must show that u in (4) is a function harmonic in KR(y), i. e., it satisfies 0. Since the integrand is twice continuously differentiable and the domain of is compact, the Laplace operator commutes with the integration 1' integral sign: = (&i)1 j — Ix — . Ix — for x (2.3.5)
  33. 33. 2.3 The Mean Value Property and Maximum PrInciple 21 According to Exercise 2.2.4 there exists a fundamental solution i'(x, such that 8 8 E Ix — f',x E KR(y). (2.3.6) From =0 and (5), one infers that = =0. (b) The expression (4) defines u(x) at first only for x KR(y). It still needs to be shown_that u has a continuous extension on KR(y) = KR(y) U I' (i.e., u E C°(KR(y))) and that the continuously extended values agree with the boundary values w: forzEf'. tim By Equation (6), putting u R2—(x—y$2 Let z e (2.3.7) 1 in Corollary 2.2.3 gives the identity f Jr foraJlxEKR(y). (2.3.8) I' be arbitrary. Due to Equation (8) one can then write: u(x) — = R2—Ix—y12J (2.3.9a) — Figure 2.3.1. ConstructIon of We define 1'O Vfl expression (9a) into ti(x) — (see Figure ço(z) R2—Ix—Y12 — = lo + I where i=40
  34. 34. 2 The Potential Equation 22 Since I — fro fro I — x— f,. it follows from Equation (8) that Jo 5 max Because of the continuity of one (2.3.9b) — F.) can choose p >0 auth that for given e >0 <€/2. Set (2.3.9c) and choose x E KR(y) sufficiently close to z such := that Ix — zf €(pfl 6(€) 1 and x — z( p/2. The last inequality implies (seeFigurel). Together with R2—lx—y12 = 2R5(€) one obtains 2RIz — xj ihI = R2—Ix—y12 f,, dl' Jr. df = Ix — d < 2C., f Jr, 1 di. and the definition of o(€) follows (2.3.9d) lId there exists a 6(€) > 0 such that Ix — zi 6(€) implies = + Iii lIol + jIg, (cf. (9c and d)). Hence, (7) has been proved, and the continuous extension of u to KR(y) leads to N Thus for every the estimate Iu(x) 0 — Exercise 2.3.13. Prove that the function u defined by the Poisson integral formula (4) belongs to and solves 2tu =0 in KR(y) even if is merely assumed to be a function integrable on F = For every point of continuity z E F of we have u(x) —' (x — z, XE KR(y)). Exercise 2.3.14. Let £1 be bounded, and let u1 and be harmonic in (1 with boundary values and on F = 0(1. Prove that: (a) on F implies u1 in (1. (b) If, furthermore, 11 is connected and if W1(x) holds for at least one point x E I' then it follows that u1 <u2 everywhere in (1.
  35. 35. 2.4 Continuous Dependence on the Boundary Data 23 2.4 Continuous Dependence on the Boundary Data Definition 2.4.1. An abstract problem of the form A(x)=y, zEX, yEY, is said to be well-posed if for all p E Y it has a unique solution x E X and if the latter depends continuously on p. It Is important to recognise whether a mathematical problem is well-posed since otherwise essential may occur in its numerical solution. In the case of the boundary value problem (1.la,b) Xc is the space of functions harmonic in S? and Y C°(f) is the set of continuous boundary data on F = oa. The topologies of X and Y are given by the supremum norm8: := sup and lIc°IIoo :=r sup (2.4.1) XEI' x€1? The question of the existence of a solution of (1.la,b) will have to be post- poned (see Section 7). The uniqueness, however, has been confirmed already in Theorem 2.3.8, if a is bounded. That the boundedness of a cannot be dropped without further ado is shown in Example 2.4.2. The functions u(x1,x2) = x1 in .17 (O,oo) x Ut u(x1, x2) = in .17 1R2K1(O) + u(x1,x2) = sin(x1)sinh(x2) in .17= (O,w) x (O,oo) and also the trivial u =0, are solutions of the boundary value problem in a, = o = aa. r (2.4.2a) (2.4.2b) (2.4.2c) =0 For bounded a the harmonic functions (solutions of (I. la,b)) depend not only continuously but also Lipschltz-continuously on the boundary data: Theorem 2.4.3. Let a o€ bounded. =0 in .0, = If ut and = and solutions of ti" on = 8.0 then Ru' (2.4.3) fly,' — PROOF. v := — U11 is a solution of v= — on 1'. According to Theorem 2.3.8, v takes its maximum and its minimum on F: —Ru" — v(x) Ru" — for all x€ Th.
  36. 36. 2 The Potential Equation 24 The definition (1) of fl 1100 implies (3). holds for a sequence of boundary values then Theorem —.0. The following — 3 shows that the associated solutions satisfy theorem states that the existence of a solution of itu = 0 in $7, ti = on F need not be assumed. If —, 0 E C°(F) be a sequence of bound0. Let tin be the ary data which converge uniforvnly toW: let — 0 in (1, solution of = S0n on F. Then the functions converge urnformly in Th to u E C2(17) fl C°(Th), and ti solves the boundary value problem Theorem 24.4. Let $7 be bouruleL Let = o a, u = on I'. - the PROOF. Since according to Theorem 3 Nun - UmIIoo converges is complete, is Cauchy convergent. Since sequence uniformly to a ti E C°(Th. According to Theorem 2.3.11, ti is harmonic (i.e. I Another problem, just as important for numerical mathematics, is rarely discussed in the literature: does the solution also depend continuously on the form of the boundary I'? Figure 1 shows domains if and 11" which approxiin Figure ic, occurs in mate $7. A polygonal domain, as shown, for the method of finite elements (see Section 8.6). Figure 2.4.1. Approximation of $7 by (2' and (2" F) := sup{dist(x, F) : x E f',,}, dist(x, F) F) 0. Here, converges one must specify when E inf{Ix y(: y E F}. C°(f). We make the definition: uniformly to C°(f',,) converges uniformly to E C°(f') if, for each e > 0, there exist numbers N(c) and 6(c) > 0 such that the following implication holds: ip, E n N(c), x€ F, y€ I',,, The sequence un E 6(e) e. (2.4.4a) C°(IZ,) converges uniformly tou E C°(.Q) if
  37. 37. 2.4 Continuous Dependence on the Boundary Data Remark 2.4.5. (a) Let K be a set which is compact (i.e. complete and E C°(K) converge C K for all n. Let bounded) with r c K, and uniformly on K to If on F then (4a) is satisfied. be the following (not continuous) Then (4b) is u on ii on (1 in the usual sense. and on C 1? hold for all n and let (b) Let = onto Th continuation of 3quivalent to uniform convergence and let 1 which are harmonic in 5Z, be solutions of Theorem 2.4.6. functions (2.4.4b) fl I?) = 0. — u(x)I: x lim 25 Let $1,., C (1, with Si -, P. Let the (2.4.5a) on F,. Then conve!ye uniformly in the sense of(4a) to — the following as8ertwfls hold: Let (a) E ifthereezistsasolutionuEC2(J?)flC°(J?) of on!' u=cp (2.4.5b) —, ti holds in the sense of (4b). then C°(a) is satisfied in the sense of (4b), then ti is —+ ti (b) if conversely the solution of (5b). PROOF. (a) Let the continuation II,, be defined as in Remark 5b. Since u is >0 for all e >0 such that there exists uniformly continuous on if lx lu(x) — Set 6(e) := exists (2.4.6a) — with 6 from (4a). Because max{Np(e), N(e/2)} (N from (4a)) we want to show that For x E S?S1,, the estimate is trivial because for x all x c a, however, there holds —u(x)I = — u(x)l —ul I' there := — u(x)l = u(x). For —+ For n for n so that (2.4.6b) It remains to estimate withn N(e)thereexistsy €1' with (cf. Theorem 2.3.8), because tin - u is harmonic in forxE — Forx€ 6(e/2). Thus we obtain — u(x)j = — u(x)l S — + — u(x)l from (4a) and (6a). Since x i',, is arbitrary it follows that ltin —til S on and (6b) proves the uniform convergence ii,, —' ti on Hence, from Remark 5b it follows that (4b) is satisfied.
  38. 38. 26 2 The Potential Equation (b) Let K C 1? be a compact set. Since 17,.. -+ 1' there exists a N(K) such N(K)} converges for n N(K). Thus, the sequence {u1,: n that K C uniformly in the usual sense on K to u so that one can apply Theorem 2.3.11: consequently u is harmonic in K. Since K C £1 may be chosen arbitrarily, it follows that u E C2(S?). By assumption, we already have U E C°(Th). That the • In Theorem 4 one was able to derive the existence of a solution t& of (5b) a as This inference is not possible for the case of just from ço, -. the following example shows. C a := K1(O){O} C 1R2. The boundaries are := Let = 0K1(O) U and V = 0K1(O) U {O}, and satisfy The —p boundary values = = 0 on 8K1(O), satisfy the condition —+ (5a) can be given explicitly: = 1 on 0) = 1 (cf. (4a) and Remark 5a). The solutions of u,..(x) = tog(IxI)/log(1/n). u(x) := 0 holds pointwise, but u = 0 satisfies neither (4b) Obviously, u,(x) nor the boundary value problem (5b). Conversely, one infers from Theorem 6a the following result: solution u a = K1(O){0} c 11t2 the potential equation has In C2(S?) fl C°(J7) which assumes the boundary values u(x) =0 on Remark 2.4.7. 8K1(O)andu(x)=linx=0. no
  39. 39. 3 The Poisson Equation 3.1 Posing the Problem The Poisson equation reads au—f in!? (3.1.la) C°(Q). In the physical interpretation f is the source term ifor example, the charge density in the case of an electrical potential uJ. To determine the solution uniquely one needs a boundary value specification, for with given f example, the Dirichiet condition (3.l.lb) DefinitIon 3.1.1. The function u is called the classical solution of the boundary value problem (la,b) if u E C2(O) fl C°(Th) satisfies the equations (la,b) pointwise. Until we introduce weak solutions in Section 7, "solution" will always mean "classical solution". The solution of the boundary value problem (la,b) will in general no longer satisfy the mean value property and the maximum principle. But these properties still hold for the differences of two solutions u, and u2 of the equation, since =0. Thus the uniqueness of the solution of — u2) = — problem (la,b) immediately follows and Theorem 2.4.3 can be brought over: f f Theorem 3.1.2. Let 0 be bounded. (a) The solution of (la,b) is uniquely deterinineii (b) ffu' and u11 are solutions of the Poisson equationfor boundary values w' and then we have IIti' — u"II,., < — (3.1.2) PROOF. (b) The proof of Theorem 2.4.3 can be repeated verbatim here. (a) If u1 and u11 are two solutions of (la,b) then (2) shows that Ilu' S R Theorems 2.4.4 and 2.4.6 can be transferred Likewise.
  40. 40. 3 The 28 Equation 3.2 Representation of the Solution by the Green Function Let the solution of (l.la,b) belong to C2(Th, where 1? is a normal domain. Then u may be as Lemma 3.2.1. u(x) = +J —! — x)] (3.2.1) for every fundamental solution in (2.2.9). The proof is the same as in Theorem 2.2.2 or in Corollary 2.2.3. The term with A = becomes Since the singularity of is integrable at = x, converges to as e —.0. -yf ExercIse 3.2.2. If(x)I integral. — (a) Let Si C ft' be bounded, xo $1, / and for 8 <n. Show that f(x)dx exists as an improper be bounded and let ii c E 5? depend continuously on with D compact. Let f(x,e) be continuous in x D with x E and let s <n. Show that F E C°(D). (b) Let D, In the boundary integral in (1) one may replace by (ci. (1.lb)). The function Ou/On on F, however, is unknown and cannot be specified arbitrarily either, since the boundary values (1.lb) already determine the solution uniquely (cf. Theorem 3.1.2). To make vanish one must select the fundamental solution so that x) =0 for e F, x €5?. DefinItion 3.2.3. A fundamental solution g In (2.2.9) is called a Green function (of the first kind) if = 0 for all €F, XE 0. The existence of a Green function is closely related to the solvability of the boundary value problem for the potential equation: Remark 3.2.4. The Green function exists if and only if for all x E $1 the boundary value problem =0 in 5? and 4i = —s(-,x) on F has a solution The above consideration results in Theorem 3.2.5. Let .0 be a normal domain. Let the boundary value problem (1.la,b) have a solutionu C2(Th. Assume the existence of a Green function of the first kind. Then one osn express u explicitly by
  41. 41. 3.2 Representation of the Solution by the Green Function u(x) = — x) L 29 (3.2.2) In the following we reverse the implication. Let the existence of the Green function be assumed. Then, does function u defined by Equation (2) represent the classical solution of the boundary value problem (1.la,b)? Here it must be proved, in particular, that u E C2(i7) and = f. Firstly, it is not even clear yet whether the function u(x) defined by Equation (2) depends continuously on x since the definition of a fundamental solution x) does not require continuity with respect to the second argument x. Despite that, the Green function is x) with respect to x in as the following result shows (cf. Leis [1, p. 671). ExercIse 3.2.6. Let a be a normal domain. Let the Green function exist for a, and for fixed y E $1 let g(., y) (weaker conditions are possible !). Now prove that g(x, y) g(y, x) for x, y E .0. (3.2.3) Hint: Apply the Green formula (2.2.5b) with = x',x" €0, u(x) := g(x,x'), v(x) :=g(x,x"), and use (2.2.10). U If one tries to reverse the assertion of Theorem 5, one encounters the surprising difficulty of having to set precise conditions on the source term 1. The natural requirement f E is necessary for u C2(fl), but it is not sufficient, as the following theorem, whose proof will be appended at the end of this section, shows. Theorem 3.2.7. Even if the boundary F and the boundary values p are sufficienLly smooth and if the Green function exists, there are functions I E C°(72) to which no solutions u Theorem 7 shows that Equation (2) need not represent a classical solu- tion for / However, a sufficient condition for / to do so is Holder continuity. Definition 3.2.8. 1 E issaid to be HOlder continuous in C(f) such that with the exponent A E (0,1) if there exists a constant C ff(x) — — yE We write I E and define the norm IIfllcA(Th) stant C which satisfies (4a): := sup { as (3.2.4a) the smallest con- (3.2.4b)
  42. 42. 3 The 30 Poisson Equation The function I E Ck(1)) is said to be k-fold Holder continuously (with the exponent A), if 171€ C"(fl) for all H k. differentiable in Here v,,0, with I"l=&'I+'"+"n (3.2.5a) is a multi-index and 17 = (3.2.5b) 1 1) a IvI-fold partial derivative operator, The k-fold HOlder continuously differon- tiable functions form the linear space Cb+A(Th) with the norm := UI k}. If 8 = k + A one also writes C'(Th) for The k-fold Lipschitz continuously differentiable functions I the choice A (3.2.4c) are the result of 1 in (4a,b). For reasons of completenesa let us add that ill k} (3.2.4d) for integer k 0. is the norm in Exercise 3.2.9. (a) f Is said to be locally Holder continuous in a if for each x a Prove that if exists a neighborhood K((x) such that f E C"(K,(x) ri Q). is compact then I E CA(fl) follows from the local HOlder continuity in Th. Formulate and prove corresponding statements for and Ck.1(Th). (b) Let 8 > 0. Show that lxi' E C'(Kft(O)), if a otherwise lxi' E Hint: 1 —t' (1—i)' for O< t 1, a 0. The function u from Equation (2) can be decomposed into u1 +u2 where and ua = — f,, çoôg/On dl'; u is the solution of the boundary UI = — f0 91 value problem (1.la,b) if we are able to show that u1 and u2 are solutions of iiuj=finf1, Theorem 3.2.10. LI the Ciw, function exists and satisfies suitable conditions then u(x) = — is j a classical solution of 4u =0 inS?, withu= x) (3.2.6) on!'. The proof goes in principle just as for Theorem 2.3.9 (cL Leis [1, p.69J).
  43. 43. 32 Representation of the Solution by the Green Function Theorem 3.2.11. Suppose the Green Junctiong(',x) exists, and let it be f E C'(Th). Then u(x) = — f 31 C2(Th{x})forx ES? J with u = I) on I'. in 0 for x E I' follows easily from 0 and (3). The property u C'(Th) and the representation result from the PROOF. The boundary condition u(x) = x be bounded and A := Exercise 3.2.12. Let (2 C E C°(A) x}. For the derivatives of / with respect to x aseume k. Prove that then with 3 <n for any and — S x) x) and IY'F(x) F(x) := H k. To prove u E C2(?)) this step cannot be repeated since x) a singularity which is not integrable. We write the derivative has in the form = (3.2.8a) — — = x3 + e and Let 8,F(x) be the difference quotient (F(xE) — F(x))/6, with = for i j. The product rule = G(x)8,F(x) + applied to Equation (8a) gives (x) = — — f (x) results in the formula (x) = — f x) is integrahie, the limit e —' — x) — 0 x) (3.2.8b) Equation (8b) implies = Jtf(E) — — so that it remains only to show that = -1. Choose KR(Z) so that x E KR(z) C (2. The Green function has the form (2.2.9): g 8 + The first two terms in f 1? x) = J OKR(5) x) are harmonic in KR(z), so that proved. + f fxR(Z) s(e, x) x) = + —1 J x) is what has to be
  44. 44. 3 The Foieson Equation 32 Let be defined by s(e, x) = — xJ) set ' 1 v(x) (cf. (2.2.1)). For fixed r > 0 (3.2.8c) / (L)flr is nonsingular For all x OKR(z) (i.e., Ix—zi r) v is harmonic, since Ofl äKr(Z) and satisfies = 0. Since s(.,x) is harmonic in Kr(Z) for r< xC, the mean-value property (2.3.1) holds, which can now be written v(x) = s(z, x) = for — Using Exercise 3.2.2b we see that v(x) is > r. — continuous in (3.2.&1) so that we also have v(x) = o'(r) for — r. (3.2.8e) Thus v is harmonic in Kr(X) with the constant boundary values (Se). The unique solution is therefore v(x) = for z — xl r. (3.2.8f) The equations (8c,d,f) yield J = x) lz 8K,.(s) and — then, since 0 < lz — xl <R, f 0 8K..(a) r n = — R'o-(R) + I$xln R2 2n - lz—x12 2n From this we see that, independently of R, n, and z, there results U From Theorems 10 and 11 follows Theorem 3.2.13. Under the same assumptions as for Theorem, 10 and 11 Equation (2) gives a representation for value problem (1.Ia,b). the classiest solution of the boundary
  45. 45. ____ 3.2 Representation of the Solution by the Green Function 33 Finally we put forward two inequalities for the Green function as exercises: Exercise 3.2.14. In (1, and Qj C Q2, respectively, let the Green functions 9, gj and 92 exist. Show (a) 0 g(x,y) s(x,y) What the inequality for n=2? for x,y E (lj C (12 for x,y E tiC n (3.2.9) 3 (3.2.10) (b) g2(x,y) Hint: Exercise 2.3.14. Supplement: Proof of Theorem 7. If we make use of a later theorem (Theorem 6.1.13) then Theorem 7 follows from not depend continuously on f, if the and that from (4d) is supremum norm (2.4.1) is used as the norm in Theorem 3.2.15. The solution u does used as the norm in C12(Th). PROOF. Let .11 K1(O) c und p =0. The disk Ills a normal region for which the Green function is known (cf. Theorem 3.3.1). By Theorem 11 there 0 on I' for the functions exist solutions of E = in (1, = p,(r) := mm p,%(r), r := IxI, 2 I r, log which belong to 00(17) and are uniformly bounded: Theorem 11 we have u"(x) = Since follows from Exercise 12 that = 1/log 2. By it = a. The first integral is bounded since where g = of the singularity function is 0) = — E C2(17). The derivative The special choice of gives for x =0 — The surface integral K := r + L — — 12jEJ5de > 0 does not depend on (0,11, so that the second integral takes on the form := K r'p0(r)dr. Since dr diverges as e —i 0, we deduce —' oo as n —. oo. Since it follows that the map I '—' u is not bounded, and thus not continuous.
  46. 46. 3 The Poiseon Equation 34 The Green Function for the Ball 3.3 Theorem 3.3.1. The Green function for the bafl KR(y) is gwen by the and 'p C2+A(r) with 0 < A < 1, the of the boundary representation formula (2.2) defines a solution u E inS?, u=ç' oni'. wJue problem function in (2.2.lia). For f E The proof of the theorem follows from a result of Schauder, which is cited in Theorem 9.1.20. in the case n = apondence (x,y) #4 can be identified with C by the corre the plane = x + iy. The following considerations are based on 2 Exerclae3.3.2. be holomorphic. Show i" = = ÷ (3.3.1) foiuEC2(S?'). Equation (1) shows, in particular, that a holomorphic transformation of coordinates maps harmonic functions into harmonic functions. An arbitrary simply connected region with at least two boundary points can, by the Riemann mapping theorem, be mapped by a conformal mapping z S? 0(z) K1(0) onto the unit disk such that = 0 for any given zo E S?. Let g(C,C') be the Green function for K1(0). One may check that G(z, zo) is again a fundamental solution. Now z ôi? im0. Thus G(z,zo) is the Green function 8K1(0), i.e., G(z,zo) = in 0. This proves plies 0(z) Theorem 3.3.3. Let Si c It2 be simply connected with at least two boundary points. Then there exists a Green function of the first kind for (1. The explicit forms of various Green functions can be found, for example, in the book by Wloka (1, Exercises 21.1—21.81. Of numerical interest might be the fact that with conformal mapping one may remove corners which are disturbing (e.g., reentrant corners) (cf. Gladwell-Wait (1, p.701). Example 3.3.4. Let 5? be the L-shaped region in Example 2.1.4. Choose = z213: $2 0'. Then # is conformal in 5?. The sides of the angle fo C 01? (cf. Fig. 2.1.1) are mapped into a single line segment, so that if has no more comersstiddngln.ThePoiasonequation4u=finflcorrespondetothe equation = in if.
  47. 47. 3.4 The Neumann Boundary Value Problem 35 3.4 The Neumann Boundary Value Problem were given on F. These so-called Dirichiet conditions or "Boundary Conditions of the First Kind" are not the only possibility. An alternative is the Neumann Condition In (1.lb) and in (2.1.lb) the boundary values u = = (3.4.1) on F. In physics this second boundary condition, as it is also called, occurs more frequently than the Dirichiet condition. For example, if u is the velocity po0 means that the gas can only move tangentially at tential of a gas, then the boundary F. Except in some unusual cases, the boundary value problem Lu = in 1? and 8u/On = 'p on F has a unique solution. An exceptional case does however occur for L = f The Poisson equation Theorem 3.4.1. Let .f? be a normal with the Neumann boundary condition (1) is only solvable if L dI't = J f(x) dx. = f (3.4.2) If a 8OhLtlOfl t& does exist, then u + c, with c any constant, is also a solution. PROOF. (1) One may repeat the proof of Lemma 2.3.6 for (2) Obviously n + c satisfies the same equation. = I. Later, in Example 7.4.8, we wifl show that the Neumann boundary value problem for the Poisson equation has a solution if and only if (2) is satisfied, and that two solutions can differ only by a constant. F,andthenormal derivative Ou/ôn occur. The Green function of the first kind was chosen in = 0 for E F. In the case of the second boundary such a way that = c (c: constant), conditions (1) one makes the assumption that &y(e, i.e. =c— for L:=j&'. = 0), from EquatIon (2) we see that must be harmonic (i.e., f cL +1 =0 is a necessary condition for the existence of Thus the condition Since on the Green Function of the Second Kind for the potential equation
  48. 48. 36 Equation 3 The Since u is only Thus the term fru&y/Ondf in (2.1) becomes const• determined up to a constant (cf. Theorem 1), one can fix this constant with the additional condition f1. u dl 0. This gives the following result, if we write g for u(x) — / + The Green function of the second kind for the ball KR(O) c JR3 can be found p. 79). in Leis 3.5 The Integral Equation Method In the representation (2.1) of the Poisson solution the singularity function a can, in particular, be chosen to be -y. If in addition one imposes the given Neumann data (4.1), one obtains u(x) dIe + g(x) for x = / k(x, a, (3.5.1) and the functions with the kernel function k(x,.) g(x) := g1(x) + := 92(x) := The right-hand side in Equation (1) with the unknown boundary value E I', can be used as an ansatz solution: + g(x). (3.5.2) = j k(x, The first suminand on the right of(2) is called the double-layer potential (dipole potential); Oi volume potential. is the single-layer potential, while 92 is a For each u C°(E') the in (2) is a solution of the equation (I.la) in 0. However, is also defined for an argument x in the exterior domain A closer look at the kernel function shows that it is in fact only weakly singular for the case of smooth boundaries F. Thus # is also defined for x E F. The function which is now defined on all is not continuous at points of the boundary I'. At XO E F there exists both an interior limit &(XO) for x xo,x E a and an exterior limit for X —' Xii, X E JR"?. In addition we have the third function value #(Xo) of (2). Their connection is given by the following jump discontinuity relation (cf. Hackbusch [7, Satz 8.2.81):
  49. 49. 3.5 The Integral Equation Method &(x0) = 2u(xo), for xo E F. + &(xo) = 37 (3.5.3a) — (3.5.3b) In order that the ansatz (2) does indeed give the solution u in (1), the boundary value &, continued from the interior, must agree with the function u which is put in the integral: & = u. Now one can solve Equation (3a,b) for &: = — i4xo). The equation & = u thus leads to u(xo) = Irk +g(xo) for F. (3.5.4) Equation (4) is called a Fredhoim integral equation of the second kind forthe unknown function u C°(fl. The original Neumann boundaryvalue problem (l.la), (4.1) and the integral eqation (4) are equivalent in the following sense: (a) If u is the solution of the Neumann boundary-value problem, then the boundary values, F, satisfy the integral equation (4). (b) If n E C°(F) is a solution of the integral equation (4), then the expreesion (1) gives a solution of (1.la), (4.1) in the entire domain il. The transformaUon of a boundary-value problem into an integral equation, and the subsequent solution of the integral equation is referred to as the integral equation method. It allows, for example, a new approach to existence statements, in that one shows the solvability of (4). The integral equation (4) can also be attacked numerically. If methods similar to the finiteelement method described in Section 8 are used, then the result is called the boundary-element method (BEM). One can find references to the integral equation method in, e.g., Hackand busch 17,
  50. 50. 4 Difference Methods for the Poisson Equation 4.1 Introduction: The One-Dimensional Case Before developing difference methods for the partial differential Poisson equa- tion, let us first recall the discretisation of ordinary differential equations. The equation a(x)1s"(x) + b(x)u'(x) ÷ c(x)u(x) = f(x) can be supplemented with initial conditions u(x1) u1,u'(x1) = *4 or with boundary conditions u(x1) = u(x2) = u2. The ordinary initial value problems correspond to the hyperbolic and parabolic initial value problems, while an ordinary boundary value problem may be viewed as an elliptic boundary value problem in one variable. In particular one can view —iil'(x)=f(x) for z€(O,1), (4.1.la) (4.l.lb) f as the one-dimensional Poisson equation = in the domain = (0,1) with Dirioblet conditions on the boundary I' = {O, 1). Difference methods are characterised by the fact that derivatives are replaced by difference quotients (divided differences), in the following called, for short, "difference? . The first derivative u'(z) can be approximated by several (so-called "first") differences, for example, by the forward or right difference: := (u(z + h) — i4z)J/h, (4.1.2a) the backward or left difference (8u)(x) := [u(x) — u(x — h)J/h, or the symmetric difference (4.1.2b) (8°IL)(x) Iu(x + h) — u(x — h)1/(2h), (4.l.2c) where h > 0 is called the step size. An obvious second difference for u"(x) is := [u(x + h) — 2u(x) + u(x — h)J/h2. One also calls The product 8, and may be viewed as 8 (4.1.3) index/ D7/emphdifference operators. or as o8, i.e.. )u(x)
  51. 51. 4.1 Introduction: The One-Dimensional Case 39 Lemma 4.1.1. Let {x—h,x-l-h)CTh. Then = u'(x) i-hR with if U E C2(Th) 8°u(x) = u'(z) + h2R with IRI (4.1.4a) if u E < (4.i.4b) = u"(x) + h2R with if u E C4(Th).(4.1.4c) PROOF. We give the proof only for (4c). If one applies Taylor's formula E h) I?.4 = u(s) ± hu'(x) + h2u"(x)/2 ± h3u"(z)/6 + h4R4, h4j zjh (4.1.5a) u"(x±t9h)/4!, (4.l.5b) with i9 E (0, 1), to Equation (3), the result is (4c) because R = u""(x — U i92h)jf24. '' '7 Slh Figura4.l.1 Gridforh=1/8 grids = {h,2h, .. .,(n — 1)h = 1— h}, (4.1.6a) (4.1.6b) of step size h = x,x ± h E only contains the values of u at 1/n. For x E STlh, Under the assumption that the solution n of Equations (la,b) belongs to C4(Th), (4c) yields the equations f(s) + 0(h2), z E ha. (4.1.7) If one neglects the remainder term 0(h2) in Equation (7), one obtains for (4.1.8a) These are n—i equations in n-fl unknowns {u,,(x), z equations are supplied by boundary conditions (ib): uh(O)=p0, Uh is a grid function defined on = The two miming (4.1.8b) Its restriction to — h))T. yields the vector
  52. 52. 40 Equation 4 Difference Methods for the and tIh(l) with the aid of If in (8a) one eliminates the components Equation (8b), one gets the system of equations (4.1.9a) with 2 —1 2 —1 —1 2 —1 Lh=h2 —1 (4.1.Db) 2 —1 —1 —1 2 f(2h), f(3h),.. . ,f(1 — 2h), 1(1 —4) + 'lb = (1(h) + (4.1 .9c) 4.2 The Five-Point Formula First we select the unit square Q={(x,y):0<z<l, O<zy<l} the fundamental domain. More general domains will be discussed in Section 4.8. In the discretiastion process is replaced by the grid as := {(x,g) for step size 4= (1:x/h, p/hE 1) (4.2.la) (n E N). The discrete boundary points form the set F5 := {(x,y) E F:x/h,y/h E Z}. (4.2.lb) As in (1.lb) we set Th5 := = {(x, y) U Z}. fl5: z/h, p/h (4.2.lc) 0 0 0 0 • • • • o •: • • • • 0 0: PorntsofFh • • • . • 0 • A two-dimensional grid 0 • Figure4.2.1 0 • V • 0 , • • • • • • • Points of
  53. 53. 4.2 The Five-Point Formula 41 In the Poisson equation =f — in 11, (4.2.2a) (4.2.2b) can each be replaced by the respective and differences (1.4c) in the x and y directions: the second derivatives = h21*L(z — h,y) +u(x + h,y)+ u(x,y — h) + u(x,y + h) — 4u(x,y)). (4.2.3) Since on the right side of (3) the function is evaluated at five points, is also called the Five- Point Formula. The discretisation of the boundary value problem (2a,b) using leads to the difference equations = 1(x) for x E forxEl'4. tLh(x)=çp(x) (4.2.4a) (4.2.4b) Through (4a7b) one obtains one equation per grid point x = (x, y) E = Except in the one-dimensional case, there exists no natural arrangement of grid points, thus one cannot immediately obtain a matrix representation as in (1.9b). The only natural indexing of is that through x 11h or the pair (i,j) Z2 with x (x,y) = (ih,jh). Let the matrix elements be given by arid hence one equation per component of the grid function uh (4h2 —h2 0 x= otherwise. (4.2.5) L, is a diagonal element; in the second case, L4 = —h2, we say that x and are neighbours. If one eliminates the components For xE with the aid of Equation (4b), then Equation (4a) assumes the following form: = qh(x) for x E ah, (4.2.6a) where := ía + fa(x) := 1(x), Wh(X) := — (4.2.6b) For the proof split the sum = Oh and .... The second partial sum is to the right side of the equation. into and is moved
  54. 54. 4 Dlfferenee Methods for the Poisson Equation 42 Remark 4.2.1. fh is the restriction of f to the grid For all points far said to be far from the boundary if all neighbours x±(0,h), x±(h,O) belong to flh. In the case of homogeneous boundary values = 0 we have = from the boundary we have ph(x) =0; here x E Qh is The system of equations (6a) can be expressed in the form (l.9a): Lhuh = the matrix where Lh = are described = by their components (cf. (5), (6b)). Strictly speaking Lh is not a matrix in the usual sense but a linear mapping since the indices x E (1h are not ordered. and the grid functions Uh = (ua(x))xEnh and Exercise 4.22. (a) An N x N matrix P is called a permutation matrix if w := Pv, for all v E IRN, has the coefficients 1 i N, where iv is a permutation of indices {1,. P-i = PT. ,N}. Show that P is unitary, i.e., (b) Let I be an index set with N elements. Let the "matrix" coefficients of indices corresponds the I, be given. To the arrangement al, /3 with = A be the matrix which NxN matrix A = belongs to a second arrangement &1,. aN of I. Prove that there exists a permutation matrix P such that A = PAPT. = ApQ for all a,/3 €1. For each arrangement of the indices the (c) Let corresponding matrix is symmetric. is lexicographical A possible linear enumeration of indices x ordering (h, h), (2h, h), (3h, h), •. ., (1 — 4, h), (4,24), (24,24), (3h,2h), •.., (I — 4,24), (4, 1 — 4), (2h, 1 — h), (3h, 1 — 4), •.., (1 — 4,1 — 4). (4.2.7) precedes the point y = (yj,.. ,yd) in Generally, the point x = . , (1,.. , d} the conditions 5d = (i > i) and lexicographical order, if for a j <y, hold. Each line in (7) corresponds to a so-called s-row in the grid 1?,,. A vector uh whose (n - 1)2 components are enumerated in the series (7) thus separates into n —1 blocks (so-called s-blocks). The block decomposition of which is given the vectors generates a block decomposition of the matrix in(8). Exercise 4.2.3. (a) With the lexicographical numhering of grid points the matrix has the form
  55. 55. 4.2 The T —I —I T Lh=h2 4 4 43 —1 —1 —1 —1 .•. ... .. —I T —1 —I ..• Five-Point Formula T 4 —1 —1 —1 4 (4.2.8) where T is an (n — 1) x (n — 1) matrix and Lh contains (n — the (n — 1) x (n — 1) identity matrix. (b) Let (1 be the rectangle 1)2 blocks. I is (1=(0,a)x(0,b)={(x,y):0<x<a,0<y<b}. Let the step size h satisfy the conditions a = nh and b = mh. Show that the leads to a matrix which also discretisation (4a,b) in the corresponding grid has the form (8). But here Lh contains (m — 1)2 blocks of the size (n — 1) x (n—i). Another frequently used arrangement is the chequer-board ordering (or red-black ordering). To this end one chessboard pattern divides into "red" and "black" fields: .Q, :={(x,y)Eflh:(x+y)/hodd}, Qb := {(x,y)E Qh:(z+y)/heven}. 429 First one numbers the red squares (z, y) E lexicographically, and then those of (Ihb The partition (9) induces a partition of vectors into 2 blocks . and a partition of the matrix Lh into 2.2=4 blocks. Exercise 4.2.4. WIth respect to the chequer-board ordering, the matrix Lh assumes the form A = II2 (4.2.10) AT 4 where, and in general, A is a rectangular block matrix because for n even, contain a different number of points. The complete (n — 1)2 x (n — 1)2 matrix in (8) or (10) is needed neither for the theoretical investigation of the system of equations LhUh = nor for its numerical solution. All properties of Lh considered in the followuig are invariant with respect to re-numbering of the grid points. Even though nuiner- ical methods for the solution of LhU,, = implicitly use an arrangement of grid points (with the exception of special algorithms for parallel computers), they never employ the complete (n — 1)2 x (ii — 1)2 matrix Every usable
  56. 56. 44 4 DIfference Methods for the Poieson Equation algorithm must take into account that Lh is sparse, i.e., it has substantially more zero than nonzero elements. In the following we again return to indexing by x Oh. Neverthelese we will continue to refer to the Lh defined by (5) as a matrix. is also described by the star The difference operator 1 = h2 1 —4 1 . (4.2.11) 1 The general definition of a difference Star (with variable coefficients) reads co.1(z,y) c_l,o(x,y) c_1,_1(z,y) = ci,i(z,y) cj,o(x, y) co,_i(x,y) c1,_1(x,y) y) y)uh(x + ih, y + jh), (4.2.12) in which the zero coefficients have not been written out. Attention. The star (11) does not represent a submatrix of Lh! The coefficients of the star appear in each row of Lh. cannot be equated Remark 4.2.5. Note that the difference operator with the matrix Lb sance 4h does not contain information on the type or place of the boundary conditions. We shall say the matrix Lb belongs toa difference star (12) if the system of equations Lhuh = results from the difference equations in x E ak after elimination of the Dirichiet boundary x "h. Even if the vector Ub in LhUh = qh contains values uh(x) = one occasionally equates ub with the grid only components Uh(X), x E function on Thh which assumes the prescribed boundary values (4b) on Ph. 4.3 M-matrices, Matrix Norms, Positive Definite Matrices The elements of the matrix A are denoted by E I. Mere A and the index set I assume the places of Lb and Slh. We writs AB and define analogously A by 0. foralla,/3€I, B,A > B,A <B. The zero matrix is denoted
  57. 57. 45 4.3 M-matrices, MatriX Norms, Po6itive Definite Matrices Definition 4.3.1. A is called an M-matrix if forallaEI, OapO OaQ>0 A nonsingular and A' (4.3.la) (4.3.lb) 0. The inequalities (is) can immediately be proved for Lh (cf. (2.5)). However we still need criteria and auxiliary results to prove (ib). I if I is said to be directly connected with The index a 0. We say that a i is connected with /3 I if there exists a "connection" (chain of direct connections) (1 i k). a =ao,al,a2,",ak =/3 with (4.3.2) The index set I together with the direct connections form the graph of A 0 holds if (cf. Fig. 1). A has a symmetrical structure, i.e., and only if 0. In this case a is (directly) connected with (3 if and only if is (directly) connected with a. Definition 4.3.2. A matrix A is said to be irreducible if every a €1 is connected with every (3 E I. 1 1 o A=011 101 FIgure 4.3.1. C) ,— ® ) @ Indices connections Graph of the irreducible matrix A In the case of the matrix A = Lh, two indices x,y are connected if and only if y = x or if y is a neighbour of x. Arbitrary x, y E can evidently . ,x(k) be connected by a chain x = = y of neighbouring points. Thus L5 Is irreducible. Exezciee 4.3.3. Prove that A is irreducible if and only if there is no ordering of the indices such that the resulting matrix has the form where A,1 and A22 are respectively square n1 x n1 and n2 x n2 matrices (iii 1,n2 1), and A12 is an n1 x submatrix.
  58. 58. 4 Difference Methods for the Poisson Equation 46 The important question as to whether A = Lh is nonsingular can be treated as a special case of the following statement. Criterion 4.3.4.__(Cerahgorin) Let Kr(z) denote the open disk {C E C: lz <r}, and let Kr(Z) := {C E C: fz — Cl r) denote the dosed disk. (a) AU eigenvalues of A lie in with r0 = >2 U (b) if A is irreducible, the elgenvalues even lie in U aEI aEf PROOF. (a) Let A be an eigenvalue of A and u a corresponding eigenvector which, without loss of generality, satisfies =1, where max{luaI: a E I} (4.3.3) is the maximum norm. There exists (at least) one = Assertion 1. 1 I with 1u71 = 1. implies >2 IA — >2 = r7. (*) From (*) follows A and hence the statement. To prove the assertion use the equation from Au = Au associated to the index -y: thatis 'El From lts.,I= 1 follows IA—a.rI By taking the modulus into the sum and by using lu,l = 1, (*) follows. (b) Let A be irreducible and let A be an arbitrary elgenvalue of A with associated eigenvector u which in turn is again normalised by hulk0 =1. The A E UQEI immediately iee.ds to the statement. Therefore let A be assumed. Assertion 2. Let and IA — a.rvl = 0, i.e., implies is directly connected with /3; then = r0. IA — = 1 = 1 and Part (a) proves the existence of a 'y E I with According to the assumption, IA = — = 1 and IA — mtwt hold so that r7. 2 is
  59. 59. 4.3 M-matrlces, Matrix Norms, Positive Definite Matrices Since A is irreducible, for an arbitrary /3 E applicable to = /3, connection (2) of with 8: °o I 47 there exists a 0. Assertion 2 shows = = r0 and A 1 for all i =0,• Since was chosen arbitrarily, it and the statement is proved. Proof of Assertion 2. Besides the inequality chain (*) there also holds IA — = so that all the inequalities in (*) become equations. In particular in particular, A E for /3 = follows that A E L1a7p1 IupI= hulk0 = 1, the identity 0 implies satisfied for each suminand. Hence of Assertion 1 to /3 yields must hold. Since proves IA — = must be lupI = = 1. The application A Kr0(aaa) U I Exercise 4.3.5. Let : a connection (2) exists between a and /3). Show that the eigenvalues of A lie in fl U aEI fiEIa Definition 4.3.6. (a) A is said to be diagonally dominant if > <IaQOCI all a I. (b) A is said to be irreducibly diagonally dominant if A is irreducible, the inequality (4a) holds for at least one index a I and for foralla€I. (4.3.4b) Note that while an irreducible and diagonally dominant matrix is irreducibly diagonally dominant, the reverse need not hold. The matrix Lh from Section 4.2, while not diagonally dominant, is irreducibly diagonally dominant, for Lh is irreducible and satisfies (4b). At all points near the boundary—i.e., those x E ah, which have a boundary point y E rh as a neighbour—however, (4a) holds: 3h2 <4h2 The spectral radius p(A) of a matrix A is given by the eigenvalue that is largest in modulus: p(A) := ma.x{IAI: A elgenvalue of A}. (4.3.5)
  60. 60. 48 4 Difference Methods for the Ponson Equation In the following we split A into A = D — B, diag {a00:a €1), D (4.3.6a) where D is the diagonal part of A: = a00, B := D 0 for a /3. (4.3.6b) a /3. (4.3.6c) - A is the off-diagonal part: = 0, Criterion 4.3.7. = for Let (6a-c) hold. Sufficient conditions for p(D'B) < 1 (4.3.7) are the diagonal dominance or the irreducible diagonal dominance of A. PROOF. (a) The coefficients of C := D1B read C0p c00=0. =—a0j,/a00 From the diagonal dominance (4a) follows that r0 := Icosi < 1 I. By the Gershgorin Criterion 4a all elgenvalues A of C lie for all a so that IAI < maxr0 < 1 and hence in Kr,,(Cao) U0€1 p(C) = p(D'B) < 1 also follows. (b) If A is irreducibly diagonally dominant then rp 1 for all /3 I and r0 < 1 for at least one a. According to Criterion 4b all eigenvalues of C lie This set lies in K1(0) UflPE,OKr,(0)C in . K1(O). At first let us assume that all rp agree: r for all /3. Since r0 < 1 for one a E I, it follows that r < 1 and flfiôKr5(0) = OKr(O) C K1(O). But if are not equal then all aKr0(o) is empty. Thus in both cases A E holds and (7) is proved. Exercise 4.3.8. (a) Weaken irreducible diagonal dominance as follows: Let A satisfy the inequalities (4b) and for all /3 I let the connection (2) exist for an index a E I for which the strict inequality (4a) holds. Prove that even under this assumption p(D'B) < 1 holds. Hint. Use Exercise 5. (b) Show that the geometric series S = p(C) < 1. Then the following holds: S = (I the form QRQT (Q a converges if and only if Hint: Represent C in unitary, and R an upper triangular matrix) and show — C)'. K[p(C)Jt'. IICiIoo (c) Let u be a vector. We define For as the vector (f) with the entries w0 (a E I). Show that: two vectors one writes v w if v0 (1)ABO,ifA?O,BO;AB>O,ifA>O,B>O; (2) AD >0 if A >0, and D 0 is a nonsmgular diagonal matrix; (3) A's' Aw ifA Oandv w; f(vtL1, ifOvw; :=
  61. 61. 4.3 M-rnatrices, Matrix Norms, Positive Definite Matrices (4) 49 Alut if A 0. Au lAut The importance of inequality (7) results from (Ia). Let D and B be defined by (6a-c). A is Lemma 4.3.9. Let A an M-matrix if and only if p(D'B) < 1. PROOF. (a) Let C := D'B satisfy p(C) < 1. Then the geometric series S := CTM converges (cf. Exercise 8b). From D1 0 and B 0 one infers C 0, C" 0, and S 0. Since I = 5(1—C) = SD'D—B (SD)'A, A has the inverse A' = SD'. D1 0 and S 0 result in A' 0. From this (ib) also results, i.e., A is an M-matrix. (b) Let A be an M-matrix. For an eigenvalue A of D'B select an eigenvector u 0. According to Exercise 8c we have tAt Because so liii = tAut = tD'BuI < A-'D 0 (cf. (la,b)) one obtains IuI that = = (1 = A'D(I — D'B)tu) A'D)u) — — — IAI)A1DIuI 1 we would get the inequality, tul 0, i.e., u = follows. For 0, in contradiction to the assumption u elgenvalue of C = D1B, thus 0. From this follows IA) < 1 for every p(D'B) <1. U Criterion 7 and Lemma 9 imply Criterion 4.3.10. If a matrix A with the property (la) is diagonally dominant or diagonally dominant, then A is an M-matrsx. Theorem 4.3.11. An irreducible M-matrix A has an element-wise positive inverse: A' > 0. PROOF. Let a,$ E I be selected arbitrarily. There exists a connection (2): = ..,c4 =$. Set C :=r D'B. Since >0, it follows that = . . Coo Ca cc > 0. 71,,7k—jEI to Lemma 9, p(C) <1 holds, so that S := C" converges. > 0 and a, /3 E I are arbitrary, S > 0 is proved. The assertion results from A' = SD' >0 (cf. proof of Lemma 9). U According Since In the following we derive norm estimates for A'.
  62. 62. 50 4 Difference Methods for the Poisson Equation Definition 4.3.12. Let V be a linear space (vector space) over the field of real numbers (K := IR) or complex numbers (K := C). The functional called . II a norm in V if =0 only for u hiu+ vii hull + Dvii hull iiAuli hAl 111111 0, for all u,v E for all A (4.3.8a) (4.3.8b) V, K,u V. (4.3&) where #1 := is the number of elements of the index Example. Let V = set I. The maximum norm defined in (3) satisfies the norm axioms (8a-c). is called a vector norm. If one views the elements u E V as vectors, But the matrices also form a linear space. In the Latter case one calls U a matrix norm. A special class of matrix norms is contained in Definition 4.3.13. Let V be the vector space with vector norm one calls (hAul := sup{flAuhh/hluhl: 0 u V} the matrix norm associated with the vector norm fi ExercIse 4.3.14. Let (b) the following holds: luAu DiBhil, (I: unit matrix), 1111111 = 1 flAull (fl.A(fl hull, hAul p(A). fl. Then (4.3.9) . fi ifi be defined by (9). Show that: (a) IHABIII . is a norm; (4.3.lOa) (4.3.lOb) (4.3.lOc) (4.3.lOa) Example. The matrix norm associated with the maximum norm fi. (cf. (3)) is called the row sum norm and is also denoted by Ii It has the explicit representation hhAik = (4.3.11) ExercIse 4.3.15. (a) Prove (11). (b) For matrices 0 < B C there holds In the next theorem we denote by I the vector having only ones as components: L=1 For the notation v w see Exercise 8c. Theorem 4.3.16. Let A be an M-matriz and let a vector w exist with Aw 1. Then flA1fl00 <
  63. 63. 4.3 M-matrices, Matrix Norms, Positive Definite Matrices be the vector with the compo- PROOF. As in the proof of Lemma 9, let u we have lul nents obtain LA'ut 51 Since A-1 0, we UuU00A'Aw = Definition 13 implies that < (cf. Exercise 8c) and How to estimate with the aid of a majorising matrix is shown in Theorem 4.3.17. Let A and A' be M-mainces with A' A. Then the following holds 0 A'' PROOF. A'-' < A1 follows from A' — A' (4.3.12) and A'' = A'(A' A)A'' and — 0, A' — A 0, A'' 0. The remainder follows from Exercise 15b. I Exercise 4.3.18. Prove (12) under the following weaker assumptions: A is an M-matrix, A' satisfies (is) and A' A. Hint: Repeat the considerations from the first part of the proof of Lemma 9 with the matrices D' and B' associated to A'. Exercise 4.3.19. Let B a principal submatrix of A, that is, there exists a, (3 E I'. Prove a subset I' C I such that B is given by the entries bap = holds for that if A is an M-matrix, then so is B and 0 S all a, /3 I'. Hint: Apply Exercise 18 to the following matrix A': a'00 = = 0 otherwise. = (a lI'), (a,j9 I'), Another well-known vector norm is the Euclidean norm (4.3.13) 11u112 := with fixed scaling constant c> 0 (for example, the choice c = h2 in connection repwith the grid functions from Section 4.2 results in the fact that resents an approximation to the integration The matrix norm associated to •))2 is independent of the factor c. It is called the spectral norm and is also denoted by )f The name derives from the following characterisation. Exercise 4.3.20. Prove: (a) for symmetric there holds (IA))2 = p(A) (cf. (5)). (b) For each real matrix holds: 1(A))2 = (p(ATA)Ih/2 (c) For each matrix holds = (maximal eigenvalue of ATAI"2. UAII2IIAT 112 Hint: (b) and (lOd).
  64. 64. 52 4 Difference Methods for the Poisson Equation For the proof in the exercise use the scalar product (u,v) : CLUaVQ (4.3.14a) ciCI (c as in (13)) and its properties (Au,v) = (u1ATv), (u,u) = I(u,v)I IluII2tIvU2. (4.3.14b) Here the case K = Ut is always used as basis., i.e., all matrices and vectors are real. Definition 4.3.21. A matrix A is said to be positive definite if it is symmetric and (Au,u) >0 for all (4.3.15) Exercise 4.3.22. Prove that (a) a symmetric matrix is positive definite if and only if all eigenvalues are positive. (b) All principal submatrices of a positive definite matrix are positive definite (cf. Exercise 19). (c) The diagonal elements a positive definite matrix are positive. (d) A is called positive semi-definite if the inequality (15) holds with "" instead of">". A positive semidefinite matrix A has a unique positive semidefinite square root B = A"2, which has the property B2 = A. If A is positive definite, then so is A1t3. A corollary to Exercise 22a is Lemma 4.3.23. A positive definite matrix A u nonsingular and has a pos- itive definite inverse. The property "A1 is positive definite" is neither necessary nor sufficient to ensure the property "A-' 0" of an M-matrix. In both cases, however, (irreducible) diagonal dominance is a sufficient criterion (cf. Criterion 10). Criterion 4.3.24. If a symmetric matrix with positive diagonal entries is diagonally dominant or inite. diagonally dominant then it is positive def- PROOF. Since resp. the Gershgorin circles which occur in Criterion 4 do not intersect the semi-axis (—oo, 0), so that all the elgenvalues must be positive. By Exercise 22a then A is positive definite, Lemma 4.3.25. be )tmin and the smallest and largest eigenvalues of a positive definite matrix A. Then there holds UA(12 = Amex, tIA'1j2 = 1/Arnie. (4.3.16)

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