Elliptic Differential Equations: Theory and Numerical Treatment (Springer Series in Computational Mathematics, 18)

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Wolfgang Hackbusch SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS

Elliptic Differential Equations Theory and Numerical Treatment

t2J

Springer

18

Springer Series in Computational Mathematics Editorial Board R. Bank R.L. Graham J. Stoer R. Varga H. Yserentant

18

Wolfgang Hackbusch

Elliptic Differential Equations T heory and Numerical Treatment

Translated by Regine Fadiman and Patrick D.F. Ion

123

Wolfgang Hackbusch MPI für Mathematik in den Naturwissenschaften Inselstr. 22-26 04103 Leipzig, Germany [email protected]

ISSN 0179-3632 ISBN 978-3-540-54822-5 (hardcover) ISBN 978-3-642-05244-6 (softcover) DOI 10.1007/978-3-642-11490-8

e-ISBN 978-3-642-11490-8

Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010923170 Mathematics Subject Classification (2000): 35J20, 35J25, 35J30, 35J35, 35J50, 35J55, 65N06, 65N12, 65N15, 65N25, 65N30 © Springer-Verlag Berlin Heidelberg 1992, First softcover printing 2010 © B.G. Teubner, Stuttgart 1987: W. Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen. Mit Genehmigung des Verlages B.G. Teubner, Stuttgart, veranstaltete, allein autorisierte englische Ü bersetzung 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

This book has developed from lectures that the author gave for mathematics students at the Ruhr-Universitiit Bochum and the Christian-Albrechts-Universitiit Kiel. This edition is the result of the translation and correction of the German edition entitled Theorie und Numerik elliptischer Differentialgleichungen. 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.

A: Theory of elliptic equations

Elliptic boundary value problems

E:Theoryof iteration methods

B: Discretisation: Difference Methods, finite elements, etc.

t--------

-------

C: Numerical analysis convergence, stability

Discrete equations

D: Equation solution: Direct or with iteration methods

The theory of elliptic differential equations (A) is concerned with questions of existence, uniqueness, and properties of solutions. The first problem of

VI

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 (0) 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 simple problem, since from 103 to 106 unknowns can occur. 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., eigenvalue 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, is 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 meth-

ods). 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. Wittum and J. Burmeister for the help in reading and correcting the manuscript of this book. He thanks Teubner 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 (§3.5) 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

Table of Contents

Foreword . . . .

v

Table of Contents

VII

Notation

XIII

1

2

3

Partial Differential Equations and Their Classification Into Types . . . . . . . . . . . 1.1 Examples . . . . . . . . . . . . . . . . . . 1.2 Classification of Second-Order Equations into Types 1.3 Type Classification for Systems of First Order 1.4 Characteristic Properties of the Different Types

4 6 7

The Potential Equation

12

2.1 2.2 2.3 2.4

12 14 17 23

Posing the Problem Singularity Function The Mean Value Property and Maximum Principle Continuous Dependence on the Boundary Data

The Poisson Equation 3.1 3.2 3.3 3.4 3.5

4

1 1

........... .

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 . . . .

27 27 28 34 35 36

Difference Methods for the Poisson Equation

38

4.1 4.2 4.3

38 40

4.4 4.5

Introduction: The One-Dimensional Case The Five-Point Formula . . . . . . . . . M-matrices, Matrix Norms, Positive Definite Matrices . . . . . . . . . Properties of the Matrix Lh Convergence

44 53 59

vrn

Table of Contents 4.6 4.7

4.8

5

General Boundary Value Problems . . . . . . 5.1

5.2

5.3

6

Discretisations of Higher Order The Discretisation of the Neumann Boundary Value Problem . . . . 4.7.1 One-sided Difference for au/an . 4.7.2 Symmetric Difference for au/an 4.7.3 Symmetric Difference for au/an on an Offset Grid . . . . . . . 4.7.4 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 Dirichlet Boundary Value Problems for Linear Differential Equations . . 5.1.1 Posing the Problem . . . . . . . . . 5.1.2 Maximum Principle . . . . . . . . . 5.1.3 Uniqueness of the Solution and Continuous Dependence . . . . . . . . . . . . . 5.1.4 Difference Methods for the General Differential Equation of Second Order 5.1.5 Green's Function . . . . . . . . . . . General Boundary Conditions . . . . . . . . 5.2.1 Formulating the Boundary Value Problem 5.2.2 Difference Methods for General Boundary Conditions . . . . . . . . . . . . Boundary Problems of Higher Order . . . . . 5.3.1 The Biharmonic Differential Equation . . 5.3.2 General Linear Differential Equations of Order 2m 5.3.3 Discretisation of the Biharmonic Differential ..... Equation

62 65 65 70 71

72 78 78 83 85 85 85 86 87

90 95 95 95 98 103 103 104 105

'!boIs from Functional Analysis . . .

110

6.1

110 110 111 112 114 115 115 116 119

6.2

Banach Spaces and Hilbert Spaces 6.1.1 Normed Spaces 6.1.2 Operators 6.1.3 Banach Spaces 6.1.4 Hilbert Spaces Sobolev Spaces . . . 6.2.1 L2(il) . . . . 6.2.2 Hk(il) and H3(il) 6.2.3 Fourier Transformation and Hk(m.n )

Table of Contents

6.2.4 HS(n) for Real s ~ 0 6.2.5 Trace and Extension Theorems Dual Spaces . . . . . . . . . . . 6.3.1 Dual Space of a Normed Space 6.3.2 Adjoint Operators 6.3.3 Scales of Hilbert Spaces Compact Operators Bilinear Forms

122 123 130 130 132 133 135 137

Variational Formulation 7.1 Historical Remarks Equations with Homogeneous Dirichlet 7.2 Boundary Conditions . . . . . . . . Inhomogeneous Dirichlet Boundary Conditions 7.3 Natural Boundary Conditions 7.4

144

The Method of Finite Elements

161

8.1 8.2 8.3

161 167 171 171 174 178 180 182 182 185 185 190 193 193 194

6.3

6.4 6.5 7

8

IX

8.4

8.5

8.6 8.7

The Ritz-Galerkin Method .... Error Estimates Finite Elements . . . . . 8.3.1 Introduction: Linear Elements for n = (a, b) 8.3.2 Linear Elements for n c m? . . 8.3.3 Bilinear Elements for n c m.2 8.3.4 Quadratic Elements for n c m.2 8.3.5 Elements for n c m.3 . . . . 8.3.6 Handling of Side Conditions Error Estimates for Finite Element Methods 8.4.1 Hi-Estimates for Linear Elements 8.4.2 L2 and HS 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 Biharmonic Equation The Two-Dimensional Case 8.5.2.2 8.5.2.3 Estimating Errors . . . . Finite Elements for Non-Polygonal Regions Additional Remarks . . . . . 8.7.1 Non-Conformal Elements

144 145 150 152

194 195 196 196 199 199

X

Table of Contents

8.8 9

10

200 201 201 202 202 203

Regularity . . . . . . . . . . . . 9.1 Solutions of the Boundary Value Problem in H"(n), 8 > m . . . . . . . . . . 9.1.1 The Regularity Problem . . . . 9.1.2 Regularity Theorems for n = m.n 9.1.3 Regularity Theorems for n = m.+.. 9.1.4 Regularity Theorems for General n c m.n 9.1.5 Regularity for Convex Domains and Domains with Comers . . . . . . . . . . . 9.1.6 Regularity in the Interior . . . . . 9.2 Regularity Properties of Difference Equations 9.2.1 Discrete HI-Regularity 9.2.2 Consistency . . . . . 9.2.3 Optimal Error Estimates 9.2.4 m-Regularity

208

Special Differential Equations

244

10.1

244 244 246 247 247 249 251

10.2

11

8.7.2 The Trefftz Method . . . . . . . . . . . . 8.7.3 Finite-Element Methods for Singular Solutions 8.7.4 Adaptive Triangulation 8.7.5 Hierarchical Bases 8.7.6 Superconvergence . . . Properties of the Stiffness Matrix

Differential Equations with Discontinuous Coefficients 10.1.1 Formulation . . . . . . .... . 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

208 208 210 215 219 223 226 226 226 232 238 240

Eigenvalue Problems

253

11.1 11.2

253 254 254 256 260

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

264

267

Table of Contents

12

Stokes Equations 12.1 12.2

. . . . . . . . . . . . .

XI

275

Systems of Elliptic Differential Equations Variational Formulation . . . . . . . . 12.2.1 Weak Formulation of the Stokes Equations 12.2.2 Saddlepoint Problems . . . . . . . . . 12.2.3 Existence and Uniqueness of the Solution of a Saddlepoint Problem . . . . . . . . . . . 12.2.4 Solvability and Regularity of the Stokes Problem 12.2.5 A Yo-elliptic Variational Formulation of the Stokes Problem . . . . . . . . . . . . . . . . . 12.3 Mixed Finite-Element Method for the Stokes Problem 12.3.1 Finite-Element Discretisation of a Saddlepoint Problem . . . . . . . . . . . . . . . . . 12.3.2 Stability Conditions . . . . . . . . . . . . 12.3.3 Stable Finite-Element Spaces for the Stokes Problem 12.3.3.1 Stability Criterion ......... 12.3.3.2 Finite-Element Discretisations with the Bubble Function . . . . 12.3.3.3 Stable Discretisations with Linear Elements in Vh . . 12.3.3.4 Error Estimates .... Bibliography

275 278 278 279

Index . . . .

307

282 285 289 290 290 291 293 293 294 296 297 300

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 simply 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,Bj

matrices boundary differential operators (cf. (5.2.1a,b), (5.3.6» the complex numbers HOlder- and Lipschitz-continuously differentiable functions (cf. Definition 3.2.8) k-fold and infinitely continuously differentiable functions cgc>(.fJ) infinitely differentiable functions with compact supports (cf. (6.2.3» distance of the function u from the subspace VN (cf. Theorem 8.2.1) surface differentials in surface integrals dr,drx diagonal matrix with the diagonal elements db d2, ... diag {db d2, ... } a function; often the right-hand side of a differential f equation g(.,.) Green's function (cf. Section 3.2) h step size (cf. Sections 4.1 and 4.2) H"'(.fJ), HS(.fJ), HG(.fJ), Ho(.fJ) Sobolev spaces (cf. Sections 6.2.2 and 6.2.4) KR(Y) open ball about Y with radius R (cf. (2.2.7), Section 6.1.1) I identity or inclusion (cf. Sections 6.1.2, 6.1.3) L a differential operator (cf. (1.2.6» or the operator associated with a bilinear form (cf. (7.2.9')) L the stiffness matrix (cf. Section 8.1) matrix of a discrete system of equations (cf. (4.1.9a»

XIV

Notation

linear space of bounded operators from X to Y (cf. Section 6.1.2) £ 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 IR and continuous at x. Then prove that the right side of Equation (llb) converges to uo(x) for y -+ O. Hint: First show that u{x, Y) = uo(x) + J~:[uo(€) -uo(x)]exp( -{x _€)2 /(4y)) d€/v'41iY and then decompose the integral into subintegrals over [x - 8, x + 8] and (-oo,x - 8) U (x + 8, (0). As with ordinary differential equations, equations of higher order can be described by systems of first-order equations. In the following we give some examples.

Example 1.1.8. Let the pair (1.£, v) be the solution of the system 1.£:1> + Vy

= 0,

V:l>

+ u y = o.

(1.1.13)

If 1.£ and v are twice differentiable, the differentiation of (13) yields the equations 1.£:1>:1> + v:I>Y = 0, v:I>Y + U yy = 0, which together imply that 1.£:1>:1> - uyy = O. Thus 1.£ is a solution of the wave equation (7). The same can be shown for v.

4

1 Partial Differential Equations and Their Types

Example 1.1.9. (Cauchy-Riemann differential equations) If U and v satisfy the system U:r; + Vy = 0, V:r; - u y = in n em?, (1.1.14)

°

then the same consideration as in Example 8 yields that both the potential equation (6).

U

and v satisfy

Example 1.1.10. If U and v satisfy the system U:r;

+ Vy = 0,

V:r;

+ 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) V:r;:r;

+ Vyy -

Wy

U:r; +Vy

= 0,

(1. 1. 16b) (1.1.16c)

=0

and v denote the flow velocities in x- and y-directions, while w denotes the pressure.

U

1.2 Classification of Second-Order Equations into Types The general linear differential equation of second order in two variables reads

a(x, y)uzz +2b(x, y)u:J:1I + c(x, y)uyy +d(x, y)uz + e(x, y)u y + f(x, y)u + g(x, y)

(1.2.1)

= o.

Definition 1.2.1. (a) Equation (1) is said to be elliptic at (x, y) if

a(x, y)c(x, y) - b(x, y)2 > o.

(1.2.2a)

(b) Equation (1) is said to be hyperbolic at (x,y), if

a(x, y)c(x, y) - b(x, y)2 < O.

(1.2.2b)

(c) Equation (1) is said to be parabolic at (x,y) if

ac - b2 = 0

and rank

[~ ~

:]

=2

at (x, y).

(d) Equation (1) is said to be elliptic (hyperbolic, parabolic) in is elliptic (hyperbolic, parabolic) at all (x, y) E n.

(1.2.2c)

n c rn.2 if it

1.2 Classification of Second-Order Equations into Types

5

Occasionally the parabolic type is defined only by ac - b2 = O. But one would not want to call the equation Ua:a: (x, y) + Ua: (x, y) = 0 parabolic, nor indeed the purely algebraic equation u(x,y) = O. 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 than two independent variables occur, say xl, ... ,Xn . The general linear differential equation of second order in n variables x = (Xl,"', Xn) reads n

n

l : ~;(X)Ua:.a:j i,;=l

+ l:ai(X)Ua:. + a(x)u =

f(x).

(1.2.3)

i=l

Since ua:.a:j = ua:ja:. holds for twice continuously differentiable functions, one can assume that, without loss of generality, (1.2.4a) Thus, the coefficients

~;(x)

define a symmetric n x n matrix A(x) = (~;(X»~;=l

(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 eigenvalues of the matrix A(x) have the same sign (±1) (Le. 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 eigenvalues have the same sign, and rank(A(x), a(x» = n where a(x) = (al(x),···, ~(x»T. (d) Equation (3) is said to be elliptic in n c m,n if it is elliptic at all x E n. 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

6

1 Partial Differential Equations and Their Types n

L Q,;,j 8 j 8Xi8xj, 2

i,j

which contains only the highest derivatives of L, is called the principal 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 E il. The transformation 4"': il c lRn -+ il' c lRn is assumed to have a nonsingular Jacobian matrix S = 8{Pj8x E C 1 (il) at x. Prove that Eq. (3) does not change its type at x if it is written in the new coordinates = 4"'(x). Hint: the matrix A = (Q,;,j) becomes SAST after the transformation. Use Remark 4 and Sylvester's inertia theorem (cf. Gantmacher [1, Chapter X-§2]).

e

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:J;(X,

y) - A(x, y)Uv(x, y) + B(x, y)u(x, y)

= f(x,y).

(1.3.1)

Here, U = (11.1,"', um)T is a vector function, and A, Bare m xm matrices. In contrast to Section 1.2, A need not be symmetric and can have complex eigenvalues. If the eigenvalues Al, ... , Am are real, and if there exists a decomposition A = S-lDS with D = diag{Alo"', Am}, A is called real-diagonalisable. Definition 1.3.1. (a) System (1) is said to be hyperbolic at (x,y) if A(x,y) is real-diagonalisable. (b) System (1) is said to be elliptic at (x,y) if all the eigenvalues 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

=

[0 -1] -1

0

.

It is hyperbolic since A is real-diagonalisable:

11]-1[-10][11] A = [ -1 1 0 1 -1 1 .

1.4 Characteristic Properties of the Different Types

7

The Cauchy-Riemann system (1.14), which is closely connected with the potential equation (1.6), is elliptic since it has the form (1) with A=

[01 -1] 0 '

and A has the eigenvalues ±i. The system (1.15) corresponding to the (parabolic) heat equation can be described as system (1) with

[0 -1]

A= 0

0

.

The eigenvalues (Al = 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 Al1lx

+ A2Uy + Bu = f.

(1.3.2)

If Al is invertible then multiplication by All gives the form (1) with A

=

-All A2. Otherwise one has to investigate the generalised eigenvalue problem det(AAl + A2) = O. However, system (2) with singular Al cannot be elliptic, as can be seen from the following (cf. (4) with ~l = 1,~2 = 0). A generalisation of (2) to n independent variables is exhibited in the system (1.3.3) with m x m matrices Ai = Ai(X) = Ai(xl,'" ,xn ) 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 '~n) E IRn the following holds:

o=f ({l,'"

n

det(LAi(x)~i) =f O.

(1.3.4)

i=l

1.4 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.

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 = const (see Fig. 1a). In the case of the hyperbolic wave equation (1.7), uti must also be prescribed (cf. (1.9» since the equation is of second order.

(a)

"u~IJ (b)

u, uti

u, uti

Figure 1.4.1. (a) Initial conditions and (b) initial boundary conditions for hyperbolic problems

It is also sufficient to give the values u and US/ on a finite interval [Xl. X2] if u is additionally prescribed on the lateral boundaries of the domain n 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 E [Xl. X2] at time t. The function u satisfies the wave equation (1.7) with the coordinate y corresponding to time t. At the initial instant in time, t = to, the deflection u(x,O) and velocity Ut(x, 0) are given for Xl < X < X2. Under the assumption that the string is firmly clamped at the boundary points Xl and X2, one obtains the additional boundary data u(xl, t) = U(X2' t) = 0 for all t. 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 u(x, Yo) = uo(x) may be prescribed. An additional specification of Uy(x, Yo) is not possible, since uS/ex, yo) = ua:a:(x, yo) = u(j(x) is already determined by the differential equation (1.10) and by Uo.

(a)

_·B· (b)

u

u

Figure 1.4.2. (a) Initial conditions and (b) initial boundary conditions for parabolic problems

The heat equation (1.10) with the initial and boundary values

u(x, to) =uo(x) U(Xb t) =!Pl (t),

[Xl, X2], U(X2' t) = !P2(t)

in

fort>to

(1.4.1)

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 = Xl and x = X2 have the temperatures CPl(t) and CP2(t). The initial temperature distribution at time to is given by uo(x). 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.

n

Remark 1.4.1. The shaded area in Figure 1 corresponds to t > to, and in Figure 2 to y > Yo. For hyperbolic equations one can solve in the same way initial-value and initial-boundary-value problems in the domain t :5 to, whilst parabolic problems in t < to generally do not have a solution.

°

If one changes the parabolic equation Ut - U xx = to Ut + U xx = 0, the orientation is reversed: solutions exist in general only for t :5 to. 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.

(a)

(b)

U

Figure 1.4.3. Boundary value conditions for an elliptic problem

An elliptic problem with specifications such as in Figure 1b in general has no solution. Let us, for example, impose the conditions U(x, 0) = u(O,y) = u(1,y) = and Uy(x,O) = Ul(X) on the solution of the potential equation (1.6), where Ul is not infinitely often differentiable. If a continuous solution U existed in n = [0,1] x [0,1] one could develop U(x, 1) into a sine series and the following exercise shows that Ul would have to be infinitely often differentiable, in contradiction to the assumption.

°

Exercise 1.4.2. Let cP E CO [0, 1] have the Fourier expansion 00

cp(x)

= I>~vsin(v7rx). v=l

Show that: (a) the solution of the potential equation (1.6) in the square n = (0,1) x (0,1) with boundary values u(O,y) = u(x,O) = u(l,y) = and u(x, 1) = cp(x) is given by

°

10

1 Partial Differential Equations and Their Types 00

u(X, y)

= '" .:( ) sin(v7rx) sinh(v7rY). ~sm V7r

(b) For 0 ~ x ~ 1 and 0 ~ y < 1, u(x, y) is differentiable infinitely often. Hint: !(x) = Ef3vsin(V7rx) E 0 00 [0, I], if v-too lim f3vv" = 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 n = [0, I] x [0,1/7r] with the boundary values u(x,O) = u(O, y) = u(l, y) = 0 and u(x,I/7r) = sin(v7rx) for v E IN. The solution reads u(x,y) = sin(v7rx)sin(v7ry)/sinv. Although the boundary values, for all v E IN are bounded by 1, the solution in n may become arbitrarily large since sup{ 1/ sin v: v E 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 {sin v: V E 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 n c m? As stated in Example 1.1.3, u is the real part of a function holomorphic in n. Since holomorphic functions are infinitely differentiable, this property also holds for u. In the case of the parabolic heat equation (1.10) with boundary values u(x,O) = uo the solution u is given by (1.11b). For y > 0, u is infinitely differentiable. The smoothness of Uo 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(x, y) = !p(x + y) + tP(x - y), where !p and tP result directly from the initial data (1.9). Check that u is k-times differentiable if Uo is k-times and Ul is (k - I)-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 1.2.1b,c the Definitions 1.2.3b,c distinguish the role of a single variable (time) corresponding to the eigenvalue oX = 0 in parabolic equations, and to the eigenvalue with opposite sign in hyperbolic equations. The connection between the different types becomes more comprehensible if one relates the elliptic equations in the variables Xl, ... , Xn to the parabolic and hyperbolic equations in the variables Xl,"· ,xn,t. Remark 1.4.4. Let L be a differential operator (2.6) in the variables x = (Xl! ... ,xn ) 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

1.4 Characteristic Properties of the Different Types Ut

is a parabolic equation for u( x, t)

+ Lu =

0

11

(1.4.2)

= u( Xl, ••• , X n , t). In contrast

Utt

+ Lu =

0

(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) = cp(t)v(x). The results are u(x, t) = e-Atv(x) in case (2), (1.4.4) u(x, t) = e±iv'Xtv(x) in case (3), where v(x) is the solution of the elliptic eigen v al ue problem Lv=>..v.

(1.4.5)

2 The Potential Equation

2.1 Posing the Problem The potential equa.tion from Example 1.1.3 reads Llu=O innclRn ,

(2. 1.1 a)

where Ll = 8 2 18x~ + ... + 82 18x;" is the Laplace operator. In physics, Equation (la) describes the potentials; for example the electric potential when contains no electric charges, the magnetic potential for vanishing current density, the velocity potential, etc. Equation (la) is also called Laplace's equation since it was described by P. S. Laplace in his five-volume work "Mecanique Celeste" (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 [1, 1I-§1.5]).

n

In the following

n will always be a domain.

Definition 2.1.1. The region connected. 1

n c lRn is called a domain if n is open and

The existence of a second derivative of u is required only in the boundary

n,

not on

r=8n of n. For a prescription of boundary values u='(Q) and define the norm II/lIc>.(.Q) as the smallest constant C which satisfies (4a):

_._

II/lIc>.(o) .- sup

{I/(X) - l(y)l. Ix _ yl>'

. x, Y E il, x

}

:f y .

(3.2.4b)

30

3 The Poisson Equation

The function 1 E Ck(n) is said to be k-fold Holder continuously differentiable in n (with the exponent 'x), if DV 1 E CA(n) for alllvi ~ k. Here v

= (Vb' •• ,vn ) with Vi

E

Z,

Vi

~

0,

Ivi

= Vl + ... + Vn

(3.2.5a)

is a multi-index and (3.2.5b) a lvi-fold partial derivative operator. The k-fold HOlder continuously differentiable functions form the linear space ck+A(n) with the norm (3.2.4c)

= k + ,x

one also writes C 8 (n) for ck+A(n). The k-fold Lipschitz continuously differentiable functions 1 E Ck,l(n) are the result of the choice ,x = 1 in (4a, b). For reasons of completeness let us add that

If s

(3.2.4d) is the norm in Ck(n) for integer k ~ O. Exercise 3.2.9. (a) 1 is said to be locally Holder continuous in il iffor each x E n there exists a neighborhood Ke(x) such that 1 E CA(Ke(x) nil). Prove that if n is compact then 1 E CA(n) follows from the local Holder continuity in n. Formulate and prove corresponding statements for ck+A(n) and Ck,l(n). (b) Let s > O. Show that IxI8 E C 8 (KR (0», if s ¢ IN, otherwise IxI8 E C 8 - l ,l(KR (0». Hint: 1 - t 8 ~ (1 - t)B for 0 ~ t ~ 1, s ~ O. The function u from Equation (2) can be decomposed into Ul +U2 where Ul = - J(J gl cJe and U2 = - Jr r.

(3.2.8d)

Using Exercise 3.2.2b we see that vex) is continuous in IRn , so that we also have (3.2.8e) vex) = a(r) for Iz - xl = r. Thus v is harmonic in Kr(x) with the constant boundary values (8e). The unique solution is therefore

vex)

= a(r)

for Iz - xl ~ r.

(3.2.8f)

The equations (8c,d,f) yield

f

s(~, x) drt; = wnrn-1a(max{r, Iz -

xl})

18Kr(z)

and then, since 0 < Iz - xl < R,

s({,x)~=

f 1 KR(Z)

Wn

l

iz-xi

fR f

10

s({,x) dre dr

18Kr (z)

rn-1a(/z-xl)dr+wn

lR

o

rn-1a(r)dr

Iz-xl

xln

lR

rn IR -Wn rn a(lz-xD+wn-a(r) -a'(r)dr n n Iz-xl Iz-xl n Wn Rna(R) _ Wn fR rn (_ r 1- n ) dr = Wn Rna(R) + fR ::.. dr n liz-xi n Wn n liz-xi n =Wn

Iz

-

= Wn Rna(R) + R2 _ Iz - x1 2 . n 2n 2n From this we see that, independently of R, n, and z, there results

.d

f

s(~,x)~=-l.

lKR(z)

From Theorems 10 and 11 follows

•

Theorem 3.2.13. Under the same assumptions as for Theorems 10 and 11 Equation (2) gives a representation for the classical solution of the boundary value problem (l.la,b).

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 il, and ill C il2, respectively, let the Green functions 9, 91 and 92 exist. Show (a) 0:59(X,Y):5s(x,y) forx,yEilCIRn , n~3 (3.2.9) What is the inequality for n=2? (b) 91(X,y):5 92(X,y) for X,y E!h C il2 (3.2.10) 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 Theorem 3.2.15. The solution u does not depend continuously on I, il the supremum norm (2.4.1) is used as the norm in CO(n), and that from (4d) is used as the norm in C 2 (n). PROOF. Let {} = Kl(O) C IR2 und cp = o. The disk {} is a normal region for which the Green function is known (cf. Theorem 3.3.1). By Theorem 11 there exist solutions un E Q2(n) of Llun = In in il, un = 0 on r for the functions 2 _x21Pn(r), r:= lxi, In(x) = x 2 r2

I rl-

Pn(r):= min ( n· r, log '2

1)

'

which belong to CO(n) and are uniformly bounded: II/nlloo = 1jlog2. By Theorem 11 we have un(x) = 9(~, x)ln(~) de. Since I/n(~)I :5 nl~l, it follows from Exercise 12 that

In

U:1Z1 =

-fa

9z1zJn de =

-fa

ipzlzJn de

-

fa

sZlzJn de at x = 0,

where 9 = ip+s. The first integral is bounded since ip E C 2 (n). The derivative of the singularity function is SZlZl (~, 0) = (~~ - ~i)jl~14. The special choice of In gives for x = 0 U:1Z1 (0) =

-fa

ipzlzl (~, O)/n(~) de + fa[~~

The surface integral K := IaKr(O)[~~

-~~]21~1-6Pn(I~l)de.

- ~iI21~1-5 d~

> 0 does not depend on

I:

r E (0,1], so that the second integral takes on the form In := K r-1pn(r) dr. Since t[rllog(rj2)1)-1 dr diverges as f -+ 0, we deduce In -+ 00 as n -+ 00. Since lIunllc2(.a) ~ IU:1Z1 (0)1, it follows that the map I f-+ U is not bounded, and thus not continuous. •

34

3 The Poisson Equation

3.3 The Green Function for the Ball Theorem 3.3.1. The Green function for the ball KR(y) is given by the function in (2.2.1180). For f E C>'(ll) and cP E C2+>'(F) with 0 < .A < 1, the representation formula (2.2) defines a solution u E C2+>'(ll) of the boundary value problem ~u = f in n, u = cp on F. The proof of the theorem follows from a result of Schauder, which is cited in Theorem 9.1.20. In the case n = 2 the plane IR2 can be identified with GJ by the correspondence (:c, y) +-+ z = :c + iy. The following considerations are based on Exercise 3.3.2. Let the map cp: z =:c + iy En ....... be holomorphic. Show

C= e+ i11 = p(z)

E n'

(3.3.1)

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 Pzo: zEn ....... P.zo (z) E Kl (0) onto the unit disk such that PZo (zo) = 0 for any given ZO E n. Let g(C,C /) be the Green function for Kl(O). One may check that G(z, zo) := g(Pzo (z), 0) is again a fundamental solution. Now z E an implies pzo(z) E aKl(O), i.e., G(z,zo) = O. Thus G(z,zo) is the Green function in n. This proves Theorem 3.3.3. Let n c IR2 be simply connected with at least two boundary points. Then there exists a Green function of the first kind for n. The explicit forms of various Green functions can be found, for example, in the book by Wloka. [I, Exercises 21.1-21.8]. Of numerical interest might be the fact that with conformal mapping one may remove comers which are disturbing (e.g., reentrant comers) (cf. Gladwell-Wait [I, p.70]). Example 3.3.4. Let n be the L-shaped region in Example 2.1.4. Choose P = z2/3: n --+ n'. Then P is conformal in n. The sides of the angle Fo C an (cf. Fig. 2.1.1) are mapped into a single line segment, so that n' has no more comers sticking in. The Poisson equation ~u = f in n corresponds to the equation ~v(C) = ~IClf(C) in n'.

3.4 The Neumann Boundary Value Problem

35

3.4 The Neumann Boundary Value Problem In (1.1b) and in (2.1.1b) the boundary values u =

is also defined for an argument x in the exterior domain m.n\n. A closer look at the kernel function k(x,e) shows that it is in fact only weakly singular for the case of smooth boundaries r. Thus 4> is also defined for x E r. The function 4> which is now defined on all IRn is not continuous at points of the boundary r. At Xo E r there exists both an interior limit 4>_(Xo) for x -+ Xo, x E n and an exterior limit 4>+(Xo) for x -+ Xo, x E IRn\n. In addition we have the third function value 4>(Xo) of (2). Their connection is given by the following jump discontinuity relation (cf. Hackbusch [7, Satz 8.2.8]):

3.5 The Integral Equation Method

4'+(Xo) - 4'_(XO) 4'+(Xo) + 4'_(xo)

= 2u(xo), = 24'(xo),

37

(3.5.3a) for Xo E r.

(3.5.3b)

In order that the ansatz (2) does indeed give the solution u in (1), the boundary value 4'_, continued from the interior, must agree with the function u which is put in the integral: 4'_ = u. Now one can solve Equation (3a,b) for !P_: 4'_(Xo) = 4'(Xo) - u(Xo). The equation 4'_ = u thus leads to

u(Xo)

= 44'(Xo) =

l k(xo,e)u(~") dr{ +

g(Xo) for Xo

E

r.

(3.5.4)

Equation (4) is called a Fredholm integral equation of the second kind for the unknown function u E CO(r). 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, u(e), E r, satisfy the integral equation (4). (b) If u E CO (r) is a solution of the integral equation (4), then the expression (1) gives a solution of (l.la), (4.1) in the entire domain n. The transformation 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., Hackbusch [7, §§7-9] and Kress[l].

e

4 Difference Methods for the Poisson Equation

4.1 Introduction: The One-Dimensional Case Before developing difference methods for the partial differential Poisson equation, let us first recall the discretisation of ordinary differential equations. The equation a(x)u"(x) + b(x)u'(x) + c(x)u(x) = !(x) can be supplemented with initial conditions U(Xl) = Ul. U'(Xl) = u~ or with boundary conditions U(Xl) = Ul, 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 - u"(x)

u(O)

= lex)

= 'Po,

for x E (0,1), u(l) = CPI

(4.1.1a) (4.1.1b)

n

= ! in the domain = (0,1) with Dirichlet conditions on the boundary r = {O,l}. Difference methods are characterised by the fact that derivatives are replaced by difference quotients (divided differences), in the following called, for short, "differences" . The first derivative u'(x) can be approximated by several (so-called "first") differences, for example, by the forward or right difference: as the one-dimensional Poisson equation -Llu

(8+u)(x) := [u(x + h) - u(x)Jlh, the backward or left difference

(4.1.2a)

(8-u)(x) := [u(x) - u(x - h)Jlh, or the symmetric difference

(4.1.2b)

(OOu)(x) := [u(x + h) - u(x - h)l/(2h), (4.1.2c) where h > 0 is called the step size. An obvious second difference for ul/(x)

is (-8-8+u)(x) := [u(x + h) - 2u(x) +u(x - h))/h 2 •

(4.1.3)

ao,

One also calls 8+ , 8-, and 8-8+ index/ D7 / emphdifference operators. The product 8-8+ may be viewed as 8- 08+ or as a+ oa-, Le .. (a+ a- )u( x) = 8+(8-u(x».

W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, DOI 10.1007/978-3-642-11490-8_4, © Springer-Verlag Berlin Heidelberg 2010

38

4.1 Introduction: The One-Dimensional Case

Lemma 4.1.1. Let[x - h, x

+ h] en.

39

Then

a±u(x) = u'(x) + hR with

IRI ~ 4I1uIlC2(Si)

if 1.1. E C 2 (n)

(4.1.4a)

aou(x) = u'(x) + h 2 R with

IRI ~ ~lIuIlC3(.a)

if 1.1. E C 3(n)

(4.1.4b)

a-a+u(x)

= u"(x) + h 2 R with IRI ~ 112"uIC4(0) ifuEC4 (n).(4.1.4c)

PROOF. We give the proof only for (4c). If one applies Taylor's formula u(x ± h)

= u(x) ± hu'(x) + h 2 u"(x)/2 ± h3u"'(x)/6 + h4R 4,

R4 = h- 4

l

z ±h

z

u""(e)(x ± h - e)3/3!dI;. = u""(x ± ~h)/4!,

with ~ E (0,1), to Equation (3), the result is (4c) because R u""(x - ~2h)]/24.

• • • • • • •

(4.1.5a) (4.1.5b)

= [u""(x+~lh)+

•

Figure 4.1.1

• · • • · · · · · li h We replace il = (0,1) and ilh

Grid for h = 1/8

n = [0,1] by the grids

= {h, 2h,···, (n -l)h = 1- h},

(4.1.6a)

nh = {O, h, 2h, ... , 1 -

(4.1.6b)

h, I}

of step size h = l/n. For x E ilh, a-a+u(x) only contains the values of 1.1. at x, x ± h E h • Under the assumption that the solution 1.1. of Equations (la,b) belongs to C 4 (n), (4c) yields the equations

n

(4.1.7) If one neglects the remainder term O(h2) in Equation (7), one obtains - a-a+Uh(X)

== h-2 [-Uh(X - h) + 2Uh(X) - Uh(X + h)] = f(x) for x E ilh . (4.1.8a)

These are n-1 equations in n+1 unknowns {un(x),x E equations are supplied by boundary conditions (lb):

nh }. The two missing (4.1.8b)

Uh is a grid function defined on Uh

nh. Its restriction to ilh yields the vector

= (uh(h),uh(2h), ... ,Uh(1-h))T.

40

4 Difference Methods for the Poisson Equation

If in (Sa) one eliminates the components Uh(O) and uh(l) with the aid of

Equation (Sb), one gets the system of equations (4.1.9a) with 2 -1

-1

2

-1 2

-1

-1

-1

(4.1.9b)

2 -1

-1 2

qh = (f(h) + h- 2 O. •

4.4 Properties of the Matrix Lh Theorem 4.4.1. The matrix Lh (five-point formula) defined in (2.5) has the following properties: Lh is an M-matrix, Lh is positive definite,

IILhlloo ~ 8h- 2 ,

II Lhl12

(4.4.1a) (4.4.1b) (4.4.1c) (4.4.1d)

IILh"ll1oo ~ 1/8, < 8h- 2 ,

~ 8h- 2 cos2(7rh/2)

II Lh"1112 ~ ~h2 sin- 2

t 2h )

=

2!2

+ O(h2) ~ 1~'

(4.4.1e)

PROOF. (a) In Section 4.3 we already noticed that Lh is irreducibly diagonally dominant and satisfies the inequality (3.1a). By Criterion 4.3.10 then Lh is an M-matrix. (b) Since Lh is symmetric and irreducibly diagonally dominant, (lb) follows from Criterion 4.3.24. (c) That IILhlloo ~ 8h- 2 can be read from (2.5) and (3.11). To estimate Lh"l one uses Theorem 4.3.16 with w(x, y) = x(l - x)/2. Then we have Lhw ~ 1 (even that (LhW)(X, y) = 1 unless y = h and y = 1 - h) and IIwlloo ~ w(1/2, y) = 1/8. (d) The inequalities (ld,e) result from Lemma 4.3.25 and

Lemma 4.4.2. The (n _1)2 eigenvectors of Lh are u"'" (1 ~ V,j.L ~ n -1):

u""'(X, y) = sin(v7rx) sin (J-L7rY )

(4.4.2a)

The corresponding eigenvalues are l~v,j.L~n-1.

(4.4.2b)

PROOF. Let n~D be the one-dimensional grid (1.6a) and let u"(x) .sin(v7rx). For each x E nkD there holds 8-8+u"(x) = h-2 [sin(v7r(x - h)) + sin(v7r(x + h)) - 2sin(v7rx)] = 2h- 2 sin(v7rx)[cos(v7rh) -1]

since sin(v7r(x ± h)) = sin(v7rx)cos(v7rh) ± cos(v7rx)sin(v7rh). The identity 1 - cos~ = 2sin2(~/2) then implies

54

4 Difference Methods for the Poisson Equation (4.4.2c)

Let L~D be the matrix (1.9b). Note that (a-a+u)(h) also involves the boundary value u(O) which (L~Du)(h) does not; similarly (a-a+u)(1 - h) depends on u(1). However, since u(O) = sin(O) = 0 and u(1) = sin(v7r) = 0 we have L~Duv = -a-a+uv, and (2c) can be brought over:

1$v$n-1.

(4.4.2c')

The two-dimensional grid function uVP- in (2a) can be written as the (tensor) product UV(x)uP-(y). Now we have that (LhUVP-)(X, y) is equal to the sum uP-(y)(L1DuV)(x) +uV(x)(L1DuP-)(y), so that (2b) follows from (2c'). • In the sequel we want to show the analogies between the properties of the Poisson equation (2.2) and the discrete five-point formula (2.4a,b). The analogue of the mean-value property (2.3.1) is the equation 1 Uh(X,y) = '4[u h(x-h,y)+u h(x+h,y)+uh(x,y-h)+uh(x,y+h)]. (4.4.3)

From (2.3) and (2.4a) with

f

= 0 we obtain the

Remark 4.4.3. The solution Uh of the discrete potential equation (2.4a) with f = 0 satisfies Equation (3) at all grid points (x,y) E {lh. As in the continuous case the mean-value property (3) implies the maximum-minimum principle.

Remark 4.4.4. Let Uh be a non-constant solution of the discrete potential equation (2.4a) with f = O. The extrema max{ Uh(X): x E nh} and min{uh(x):x E nh} are assumed not on n h but on n.

PROOF. If Uh were maximal in (x, y) E nh, then because of Equation (3), all neighbouring points (x± h, y) and (x, y± h) would have to carry the same values. Since every pair of points can be linked by a chain of neighbouring • points, it follows that Uh = const, in contradiction to the assumption. The last proof indirectly uses the fact that Lh is irreducible. The irreducibility of Lh corresponds to the assumption in Theorem 2.3.7 that n is a domain, i.e., connected. The result of carrying over Theorems 2.4.3 and 3.1.2 reads as follows:

Theorem 4.4.5. Let ul and u~ be two solutions of (2.4a): -L1hu~ = f for different boundary values u~ = !pi (i = 1,2). Then the following holds: (4.4.4a) (4.4.4b)

4.4 Properties of the Matrix Lh

55

PROOF. Let wh:= u~ -uk. (1) In the case that 0 (cf. Theorem

4.3.11).

•

The bound gh(X,~) :::; h- 2 /8 is too pessimistic and can be improved considerably.

Remark 4.4.8'. 0< h(X~) < 9

,

! _ 10g(lx -

- 4

~12 + 2h2) < !log hi.

log 16

-

log 4

(4.4.7)

This estimate via 0(1 log hI) reflects the logarithmic singularity of the singularity ftmction s(x,~) = -log(lx - ~12)/41T. Exercise 4.4.9. For the proof of inequality (7) define Sh(X,~) := 1/4 -log(lx - ~12 + 2h2)/ log 16,

'Uh(X):= Sh(X, 0)

and carry out the following steps: (a) Sh(X,~) ~ 0 for all x E h, ~ E n h, (b) -Llh'Uh(O) = h- 2 , (c) -Llh'Uh(X) ~ 0 for all x E }R2 (longer calculation I), (d) -Llh(Sh(-,~) - gh(·,e» ~ 0 for fixed ~ E nh, (e) 9h(X,e):::; Sh(X,e).

n

Let 0 as well as (Ck.l(D), 1I·lb.,lCD» are complete, thus Banach spaces. PROOF of (b). If {Un} is Cauchy convergent then there exists a limit u* (x) := limn--+ooUn(x) for all xED. Since Un converges uniformly to u·, u· must be continuous, i.e., u* E CO(D). •

Denote by Loo(D) the set of functions that are bounded and locally integrable on D. Here we do not distinguish between functions which agree almost everywhere. In this case the supremum norm is defined by IIuliLOOCD) := inf{sup{lu(X) I: X E D\A}:A set of measure zero}}

(Loo(D), II 'IILooCD» is a Banach space. Exercise 6.1.8. Let X be a normed space and Y a Banach space. Show that L(X, Y) is a Banach space. Exercise 6.1.9. Let X be a Banach space and Z C X a closed subspace. The quotient space XI Z has as elements the classes x = {x + z: z E Z}. Show that X/Z with the norm IIxil = inf{IIx + zllx: z E Z} is a Banach space.

A set A is said to be dense in (X, II· IIx) if A c X and A = X, i.e., every x E X is the limit of a sequence au E A. If (X, II· IIx) is a normed, but not complete, space, one calls (X, 1I·lIx) the completion of X, if X is dense in X and IIxllx = IIxlix for all x EX. The completion is uniquely determined up to isomorphism, and can be constructed in the same way as one constructs m. as the completion of 0, 0 ~ k

PROOF. (a) Partial integration for

< m, u E HO(ll).

and 7](10)

=

(6.2.lOa) (6.2. lOb)

lal = 1 shows that

IDOtul~ = (DOtu, DOtu)o = _(D 2Ot u, u)o ~ ID 20t ui0lui0 ~ lul2lulo.

Since also lul~ ~ lul2lulo, it follows that lul~ ~ (n + 1)luI2Iulo. If one replaces u by DP u with 1.81 = I, one obtains in the same way lul,2 ~ Clull+llub-l

(1 ~ l

< m; u

E

Ho(ll».

(6.2.lOc)

(b) Let u =I 0 be fixed. Set VI := log lub, WI := [lv m + {m -1)vo1/m and ZI := V, - W,. (lOc) leads to 2z1 - ZI-l - ZI+l ~ e:= loge, where Zo = Zm = O. For z = {Zl.··., zm_d T one obtains Az ~ en. Here A is the M-matrix (4.1.9b) for h = 1. A -1 2:: 0 proves that z ~ c:= eA- 1 n. VI = ZI+WI ~ CI+WI (I ~ 1 < m) implies lull = exp(vl) ~ exp(ci + WI), i.e., (lOa) with C:= exp(ellA -111100)' (c) Elementary calculation shows that for each E > 0, 0 ~ ~ eo < 1 there exists an 7]{ 10, 8 0 ) such that

e

aeb l - e ~ Ea+7](E)b for all a,b 2:: O. Formula (lOb) is a corollary of (lOa,d).

(6.2.1Od)

•

6.2 Sobolev Spaces

119

By similar means, together with (5.3.10), one proves

{DOtu,DPu)o :;; €Iul~ + {4€)-1Iul~ (€ l> 0, lal :;; m, I\PII:;; k:;; m, '11. E Hm{n)), {DOtu,DPu)o :;; €Iul~ +ll{€)lul~ (€ > 0, lal:;; m, IPI < m, '11. E H[{'(n)).

Remark 6.2.14. The set {u E COO{n):supp{u) compact, 1'11.1", in H"'{il). .

(6.2.10e) (6.2.lOf)

< co} is dense

PROOF. Let '11. E H"'(n), € > O. According to Lemma 7 there exists a function COO{n) with 1'11. - uEI", :;; €/2. There exists a E coo{m,n) with a{x) = 1 for Ixl :;; 1, a{x) = 0 for Ixl ~ 2. For sufficiently large R, one also has luE{x)-a{x/ R)uE{x)l", :;; €/2. Thus there exists v{x) = a{x/R)uE{x) E c8"{n)

UEE

with 1'11. -

vi", :;; €.

•

Since "supp(u) compact" already implies "supp(u)

cc m,n"

we obtain

The Leibniz rule for derivatives of products proves

Theorem 6.2.16. E H"'(n).

l\auI\Hk(SJ):;; C",l\al\ck(.i'i)lIuIlHk(SJ) for all a

E

C"'(Q),

'11.

Theorem 16 together with the substitution rule for volume integrals shows

Theorem 6.2.17. (Transformation theorem). LetT:n -+ n ' be a oneto-one mapping onto n ' with T E cmax(""l)(n) and IdetdT/dxl ~ 0 > 0 in n. We write v = '11. 0 T for vex) = u(T(x)). Then '11. E H"'(n' ) ['11. E H~(n')] also implies '11. 0 T E H"'{n) [E H~(n)l and (6.2.11)

6.2.3 Fourier 'Iransformation and H"'{m,n) For

'11.

E C8"(rn.n ) one defines the Fourier-transformed function

u by (6.2.12)

Note that bounded.

u is

described by a proper integral since the support of

'11.

is

120

6 Tools from Functional Analysis

Lemma 6.2.18. Let

U

E Co:'(ffin). For R

~ 00

converyes uniformly to u(y) on supp(u).

PROOF. It suffices to discuss the case n tion with respect to { results in

IR(u; y)

= (I/,K)

= 1 (by Fubini's theorem). Integra-

L

(x - y)-l sin(R(x - y»u(x) dx.

Then IR(I;y) = (1/'n") IRt-1 sin(t) dt = 1 for all R > O. Since U E coo(ffin), then also w(x, y) := [u(x)-u(y»)/(x-y) E coo(ffi2n). The estimates w(x, y) = O(I/lxl) and wa:(x, y) = 0(I/x2 ) hold uniformly for y E supp(u). Partial integration yields

IR(u(,) - u(y);y) = IR«' - y)w(·, y); y) = (1/7r) fIR sin(R(x - y»w(x,y) dx

=

-(I/7r)

L

cos(R(x -y»Wa:(x,y)dx/R = O(I/R).

The statement follows from

IR(u; y)

= U(y)IR(I;y) + IR(u(,) -

u(y);y)

= u(y) + O(I/R).

•

PROOF. Lemma 18 shows

Lemma 6.2.20. The inverse Fourier transformation 9="- l u = u is defined for u E Co:'(ffin) by (13):

(9="-lu)(x) := (27r)-n/2 {

1IR"

ei({,x)u({)d{

:= lim (27r)-n/2 [ R-+oo

11{loo~R

ei({,x)u({)d{.

(6.2.13)

6.2 Sobolev Spaces

121

PROOF. We have from Lemma 18

f

(21T)-n/2

f

ei({,x)u(e)£le = (21T)-n

J 1{loo5,.R

f

ei({.x-y)u(y) dy£le

J 1{loo5,.R JJRR

= u(x) +O(1/R).

•

Theorem 6.2.21. !1',!1'-1 E L(L2(IRn), L2(IRn» with 1I!1'1I£2(JRR)t-L2(JRR) = 1I!1'-1IlL2(IRR)t-L2(RR) = 1, i.e., !1' is an isometric mapping of L2(IRn) onto itself The scalar product satisfies (u,v)o = (u,v)o for all u,v E L2(IRn). PROOF. Since Cg:'(IRn) is dense in L2(IRn) (cf. Lemma 2), !1' can be continued to!1': L2(IRn) ..... L2(IRn) (cf. Theorem 6.1.11). The norm estimate follows from Lemma 19. The roles of !1' and :r- 1 are interchangeable (cf. (12) and (13»; thus !1'-1 E L(L2(IRn), L2(IRn» also holds. The second statement results from

(u, v)o =

~(Iu + vl5 -lul5 -lvI5) = ~(Iu + vl5 -lul5 -lvI5) = (u, v)o . •

Exercise 6.2.22. Prove that: (a) With e a

= ell .. ·e~R, there holds (6.2.14)

(b) There exists C for all e E IRn.

= C(k) such that iJ(1+leI 2)k ~ Llal91eal2

~ C(1+leI 2)k

Lemma 6.2.23. (a) lulk = h/LaI9IeaI2u(e)lo holds for all u E Hk(IRn). (b) A norm on Hk(IRn) equivalent to 1·lk is (6.2.15)

PROOF. (a) It suffices to show the statement for u E Cg:'(IRn) (cf. Corollary 15, Theorem 6.1.11): 2

lul~ =

E lal9

IDau l5 =

E lal9

I!1'D au15 =

E

leau (e)15

lal9

(b) The statement follows from Exercise 22b.

=

E lal9

lea I2u(e)

.

o

•

Lemma 6.2.24. Let 8h.j be the difference operator8h•j u(x) := [u(x+~ej)­ u(x- ~ej)}fh, where ej is jth unit vector. Ifu E Hk(IRn) and 18h,julk ~ C for all h > 0, 1 ~ j ~ n, then u E Hk+1(IRn) holds. Conversely, 18h•j ulk ~ lulk+l holds for all u E Hk+1(IRn), h > o.

122

6 Tools from Functional Analysis

PROOF. (a) From !f(u(· + 8ej))(e) = e-i6~iu(e) follows !f(8h ,ju)(e) = sin(ejhI2)u(e). Hence, since 4h- 2sin2(ejhI2) ~ ej for h ~ 1/1el, follows 1!f(8h ,ju)(e)1 2 = ejlu(e)l2 for lei ~ 1/h. Summation over j = 1, ... , n and integration over then gives

¥

e

r .. (1 + leI 2)k+1Iu(e)1 2de 1m.

(lul~+d2 =

~ ~

r ... + (lul~)2 + 11~19/h { (1 + lel 2)klel 2lul 2de

11~1?1/h

1

1~I?l/h

~ (

11~1?1/h

The inteliral over

A2 A2 ... + (Iulk) +~ .LJ(18h ,j u lk) j=l

'" + Cklul~ + nC2 •

lei ~ 1/h vanishes for

(b) For the converse use (8h ,ju)(x)

=

h -+ 0 so that the statement follows. f~~~2u:l)i(x+thej)dt. •

n

Let 8 ~ O. For = m.n one can define the following scalar product (16a) and the Sobolev norm (16b) for all u e C8"(m.n): (6.2.16a) (6.2.16b) The completion in L2(m.n) also defines the Sobolev space HB(m.n) for noninteger order 8. On the basis of Lemma 23 and Exercise 6.1.12 the newly defined HB(m.n ) for 8 e IN U {O} agree with the Sobolev spaces used until this point. Let c m.n . The number 8 ~ 0 can be decomposed as s = k + A with k e INu {O} and 0 < A < 1. We define

n

(u,v)s:=

~ lal::>k

+

lul s

Jlxn

[1

Dau(x)Dav(x)dx

n

[Dau(x) -

D~:(~)~[I~:~(X) -

:= IIUIiHaCS1) := J(u,u)s

(8 = k +A, 0

Dav(y)) dxdY] , (6.2.17a)

< 8 < 1).

(6.2.17b)

The norm I·IB is called the Sobolev-Slobodeckir norm. One can define the Hilbert spaces H8(il) and H3(n) in the same way as in the case 8 = k E IN. The properties of these spaces are summarised in the following theorem (cf. Adams [1), Wloka [1]):

6.2 Sobolev Spaces

123

Theorem 6.2.25. Let s ~ O. (a) For n = IRn the norms (I6b) and (I7b) are equivalent, i.e., both norms define the same space HB(IRn) = H3(IRn). (b) {u E COO(n): supp(u) compact, lulB < oo} is dense in HB(n). (c) CO"(n) is dense in H3(n). (d) aDa(bu) E Hs-Ial(n), if lal ~ s, u E HB(n), a E ct-ial(n), bE Ct(n), where t = s E lN U {O} or t > s. (e) HB(n) c Ht(n), H&(n) c H&(n) for s ~ t. (f) In (IOa,b) k and m can be real. (g) In (11) replace IITllc"Co) by IITllc"co)' with t > k if k rt IN. Exercise 6.2.26. Check, using the norm (I6b), as well as (I7b), that the characteristic function u(x) = 1 in [-1,1], u(x) = 0 for Ixl > 1 belongs to HB(IR) if and only if 0 ~ s < 1/2.

6.2.5 Trace and Extension Theorems

The nature of boundary value problems requires that one can form boundary values ul r (restriction of u to r = an, or trace of u on r = an) in a meaningful way. As can be seen easily, a Holder-continuous function u E CS (n) has a restriction ul r E CB(r) if only r is sufficiently smooth. But, from u E HS(r) does not necessarily follow ul r E HS(r). Since the equality u = v on HS (.0) only means that u(x) = vex) almost everywhere in .0, and r is a set of measure zero, u(x) =f vex) may hold everywhere on r. Also, the boundary value u(x) (x E r) cannot be defined by a continuous extension either since, for example, u E Hl(n) need not be continuous (cf. Exercise 4c). The inverse problem for the definition of ul r is extension: does there exist, for a given boundary value r.p on r, a function u E HS(n) such that r.p = ul r? If the answer is negative there exists no solution u E HB (.0) to the Dirichlet boundary value problem. First we study these problems on the half-space

n=

IR~ := {(Xl.··· ,xn ) E IRn:x n

Functions u E H8(IR~) restricted to r.

r = an =

IRn- 1 x {O}. (6.2.18) will first be continued to it E HB(IRn) , and then it

> O} with

Theorem 6.2.27. Let s ~ O. There exists an extension operator ifJB E L(HS(IR~), HS(IRn» such that for all u E HS(IR~) the extension it = ifJsu and u coincide on IR~. PROOF. For s = 0, set it = u on IR~ and it = 0 otherwise. Since lIuIlL2(Rn) = lIuIlL2(R+), the mapping defined by ifJou := it is bounded: lIifJolIL2(Rn) 1/2. The second is U(e') := 1I0(e', ')IIL2(IR) E L2(Rn-l) because IIUIIL2(Rn-1) = liOIIL2(Rn) (Fubini's theorem). Together we have:

Integration over e' E

m.n- 1 results in

6.2 Sobolev Spaces

125

Thus Iwl:-l/2 ::; C~lul: is proved with C~ = (Ks/(27r))1/2 (cf. (16b)). In the case n = 1, w(e') already represents 'Yu = u(O), and the integration over e' is not required. • Theorem 28 describes the restriction u(·, 0) = 'YU to Xn = o. Evidently, similarly we have lu(.,xn )ls-l/2 ::; Cslul s for any other Xn e m., with the same constant Cs. The mapping Xn 1-+ u(·,x n ) is continuous [resp. Holdercontinuous] in the following sense.

Theorem 6.2.29. For s

> 1/2 the following

statements hold:

lim lIu(·,x n ) - u(·,Yn)IIH.-l/2(lRn-1) = 0 for all Xn

'IIn--t:r:n

e IR,u e HS(IRn), (6.2.21a) (6.2.21b)

IIU(·,X n ) -U(-,Yn)llH8-1/2->'(lRn-1)::; Ks.~lxn -Ynl~IIUIlH8(lRn) for all u e HS(m.n ), 0::; >. < 1, >.::; s -1/2 (and for>. = 1 if s > 3/2).

PROOF. (a) Let Uv e C~(IRn) be a sequence with Uv -+ U e HS(IRn) and set IPv(x):= lIuv(-,x)IIHo-l/2(lRn-1). The function IPv is continuous in m. and converges uniformly to lIu(.,x)IIH.-l/2(lRn-1) since luv(-,x) - u(.,X)ls-l/2 ::; Cslu v - uls for all x e IlL Thus (21a) follows. (b) ue(·,xn ) := u(·,xn + €) - u(·,xn ) has the Fourier transform ue(e) = ue(e',en) = [exp(ien€) - l]u(e) so that lue (e)1 2 = 4sin2(en€/2)lu(e)1 2 . As in the proof for Theorem 28 set W(.,x n ) := :7n-1Ue(·,X n ), W = W(.,O). The first integral in the estimate of 27rlw(OI2 now reads

1m.(1 + 1e'12 + e;)-S sin2(en€/2) den = (1 + le'1 2) 1/2-s fIR (1 + e)-S sin2(1]t) dt with 1] = ~(1 + le'1 2)1/2. The decomposition of the last integral into subintegrals over It I ::; 1/1] and It I ~ 1/1] shows flR(1 + t 2)-S sin2 (1]t) dt ::; Cs.~1]2~. The remainder of the argument follows the same lines as in the proof of The• orem 28. Up to this point we have obtained HS(IRn) by completion of CQ'(IRn) in L2(m.n). The next theorem shows that for sufficiently large s one can also complete in CO(IRn) n L2(IRn) so that HS(IRn) contains only classical functions (Le. continuous, HOlder-continuous, [HOlder-] continuously differentiable functions).

Theorem 6.2.30. (Sobolev's lemma) Hs(m.n) c Ck(IRn) holds for k IN U {O}, s > k + n/2 and HB(IRn) c ct(m.n) for 0 < t ¢. IN, s ~ t + n/2.

e

PROOF. (a) Let s ~ t + n/2, 0 < t < 1, u e HS(IRn). For given x, y e IRn we want to show lu(x) - u(y)1 ::; Clx - ylt with C independent of x,y. The coordinates of the m.n can be rotated so that x = (Xl! 0, .. ·,0), y = (Yl. 0, ... ,0). An (n-l)-fold application of Theorem 28 to u( .), u(-, 0), u(·, 0,0)

126

6 Tools from Functional Analysis

, etc., results in 1£(·,0,0, .. ·,0) E HB-(n-l)/2(1R). Theorem 29 provides the desired estimate with C = KB-(n-l)/2,tlU(" 0,···, 0)IB-(n-l)/2 :5 C'luls ; thus 1£ E ct(lRn) and II 'IICt(Rn) :5 CII ·IIHo(Rn). Then 1£ E CO(lRn) follows from Ct(lRn) c CO(lRn). (b) Let s ~ t + n/2, 1 < t < 2, 1£ E HB(lRn). Part (a) is applicable to Datu E Hs-l(lRn) (cf. Theorem 25d,e) with lal :5 1: Datu E Ct-l(lRn ) for lal :5 1. Thus 1£ E Ct(lRn), etc. • We return to the statements of Theorems 27 and 28. For all 1£ E C8"(IR+') the restriction 'Yu = 1£(·,0) agrees with 'Y~BU. Completion in HS(IR+') yields

This proves the following corollary: Corollary 6.2.31. Let s > 1/2. For the restriction 'Yu := 1£(.,0) we have 'Y E L(H"(IR+.), H,,-1/2(lRn- 1

».

°

With the restriction to Xn = one evidently loses half an order of differentiability. Conversely one gains half an order if one continues w E Hs- 1/ 2 (lRn - 1 ) suitably in IRn. Theorem 6.2.32. Let s > 1/2, wE Hs-l/ 2(lRn - l ). There exists a function 1£ E H"(lRn ) [1£ E H"(IR+.») such that luis :5 C"lwl,,-1/2 and 'YU = w, i.e., W = 1£(.,0).

PROOF. Let u = :1'nu and w= :1'n-lw be the Fourier transforms. 'YU = w is equivalent to (20). For

u(e) = u(e', en) := w(e')(1 + le'1 2),,-1/2(1 + le'I 2 + e;)-" / K", K,,:=

L+ (1

t 2 )-"dt,

one checks that (20) and lui: = K;l/2Iwl:_l/2 hold. Restriction of u E HB(lRn) to IR+. proves the parenthetical addition. • If one replaces IR+. with a general domain n c IRn , then IRn- 1 ~ IRn- 1 x {O} = aIR+. becomes r = an, and the necessity arises of defining the Sobolev space H" (r). We begin with the

°

Definition 6.2.33. Let < t E IRU{ oo} [resp. k E 1NU{O}]. We write n E C t [n E Ck,l], if for every x E r := an there exists a neighbourhood U C IRn such that there exists a bijective mapping O}, cp(U n (llln\il» = {e E Kl(O): en

(6.2.22a) (6.2.22b) (6.2.22c) (6.2.22d)

< O}.

Here Kl(O) is a circle (ball) if 1·1 is the Euclidean norm. For the maximum norm I . 100, Kl(O) is a square (cube). Likewise, Kl(O) can be replaced by another circle KR(z) or any rectangle (xl,x1) x··· x (x~,x~).

Figure 6.2.1. Covering neighbourhoods of rand

[J

Example 6.2.34. Let il be the circle Kl(O) C lll2. A neighbourhood ofx* = (1,0) is Ul from Fig. 1. The mapping x E Ul 1-+ cp(x) := e E (-1,1) x (-1,1) with Xl = (1 - e2/2)cos(7rel/2), X2 = (1 - e2/2)sin(7ret/2) is bijective and satisfies (22a-d) with t = 00. The same holds for any x E r. Thus il E Coo. Exercise 6.2.35. (a) The rectangle il = (xl, x1) x (x2, x~) and the L-domain from Example 2.1.4 are domains il E CO· l . (b) The cut circle in Figure 5.2.1b does not belong to CO· l . Lemma 6.2.36. Let il E C t [il E C k •l ] be a bounded domain. Then there exists N E lN, Ut (O:5 i :5 N), Ut , ~ (1 :5 i :5 N) with N

U i open, bounded (0 :5 i :5 N),

U Ui :::> 'ii,

UO CC il,

(6.2.23a)

i=O

N

Ui ;=

ui n r

(1:5 i :5 N),

UU = r i

(6.2.23b)

= 1, ... , N,

(6.2.23c)

i=l

at: Ui ~

0

-4

ai(Ui ) C llln-l bijective for i

ajl E Ct(aj(Ui n Uj

»[resp. at

0

ajl E Ck.l(aj(Ui n Uj))]. (6.2.23d)

On Ui (1 :5 i :5 N) are defined mappings CPi with the properties (22a-d).

128

6 Tools from Functional Analysis

PROOF. For every x E r there exist U = U(x) and

1. The Sobolev space HB(r) is defined as the set of all functions u: r -+ m. such that (O'iU) 0 a;l E H&(m.n - 1) (1 ~ i ~ N). Theorem 6.2.39. (a) HB(r) is a Hilbert space with the scalar product N

(U,V)B:= (U,V)HO(r):= L«UO'i) o a; 1, (vO'.)oa;1)HO(lRn -

1) •

• =1

(b) If {(Ui , a.): 1 ~ i ~ N} is another C t _ [Ok,1:] coordinate system of rand {Ui} another partition of unity, then the space HB(r) defined by this is equal to HB{r) as a set. The n017nS of H8(r) and fI8(r) are equivalent.

6.2 Sobolev Spaces

129

PROOF. For (b): Transformation Theorem 17 [resp. 25g] ; for DE CO,l, cf. Wloka [1, Lemma 4.5]. • The trace and extension theorems (Corollary 31 and Theorem 32) can be extended to any domain with a sufficiently smooth boundary. "f now denotes the restriction to r = an: "fU = u Ir. Theorem 6.2.40. Let n E C t with 1/2 < s :::; t E IN or 1/2 < s < t [resp. DE e k ,l, 1/2 < s = k + 1 E IN]. (a) The restriction "fU of u E HII(n) belongs to Hs- 1/ 2(r): "f E L(HS(D), HS-l/2(r)). (b) For each W E Hs-l/2(r) there exists an extension u E HS(D) with W = "fU, luis:::; e s lwls-l/2. (c) For each W E HS(n) there exists a continuation Ew E H8(ntn) with E E L(H8 (n), Hs(ntn)). PROOF. The proofs follow the same pattern. Let U i , Ut and O. Set 8(rJ) := IIv'IIL3(o.'1) and note that 8(rJ) -+ 0 for rJ -+ O. Since vex) = v'l(x) for x 2:: rJ, it remains to estimate the variables IIv - v'lIlL3(o.'1) and IIv' - v~IIL3(o.fj)' Because of v~ - v' = cP~v + (cp'l - l)v', one obtains IIv~ - v'IIL3(o.'1) :::; IIcp~IILoo(o.'1)lIvIlL3(o.'1)

:::; (3/rJ)lI v IlL3(o.'1)

+ IIcp'l -

lIlLOO(o.'1)lI v 'IIL2(o.'1)

+ 8(rJ)·

J;

Since vex) = v'(e) cIe, (5b) implies the estimate Iv(x)1 :::; .jij8(rJ) for all o :::; x :::; rJ, and hence IIvIlL3(o.'1) :::; rJ8(rJ). Then the statement follows from Iv - v'll~ :::; IIv - V'llli3(o.'1) + IIv' - v~lIi3(o.'1) :5 C8 2 (rJ) -+ O. • When n E C1, the normal direction n exists at all boundary points. Analogously to Theorem 42 one proves Corollary 6.2.43. For n E C1 and k E IN holds H~(n)

= {u E Hk(n):o'u/on'lr = 0 forO:::; 1:5 k -I} = {u E Hk(Il):DQul r = 0 forO:::; lal:5 k -I}.

6.3 Dual Spaces 6.3.1 Dual Space of a Normed Space Let X be a normed, linear space over ffi.. As a dual space, X' denotes the space of all bounded, linear mappings of X onto ffi.: X':= L(X,ffi.).

6.3 Dual Spaces

131

According to Exercise 6.1.8, X' is a Banach space with the norm (dual norm)

Ilx'lIxl ;= IIx'IIR t) are compact. (b) FUrther, let n E CO,l. The embeddings Hk(n) c H'(n) (k, lEN U {O}, k > 1) are compact. (c) Let 0 :::;; t < sand n E or (r > t, r > 1) or n E Ok,l (k + 1 > t). Then the embedding H8(n) c Ht(n) is compact. Remark 6.4.9. In Theorem 8b one can replace n E 0°,1 by the "uniform cone property" (cf. Wloka [1, §2.1]). To ensure n E 0°,1 it is sufficient that the boundary an is piecewise smooth and the inside angles of possible comers are smaller than 211'. Indented comers (cf. Figure 2.1.1) are thus permitted while a cut domain (cf. Figure 5.2.1b) is excluded. In Section 6.S the following situation will arise:

v cUe V'

Gelfand triple,

T E L(V', V).

(6.4.1)

Because of the continuous embeddings, T also belongs to L(V', V'), L(U, U), L(V, V), and L(U, V).

Theorem 6.4.10. Let (1) hold. Let V CUbe a compact embedding. Then T E L(V', V'), T E L(U, U), T E L(V, V), T E L(V', U), and T E L(U, V) are compact. PROOF. As an example, let us do T E L(U, V). Since the inclusion I E L(V, U) is compact we see I E L(U, V') is also compact (cf. Lemma Sb). T E L(U, V), as the product of the compact mapping I E L(U, V') with • T E L(V', V), is compact (cf. Lemma 5a).

Exercise 6.4.11. Let dim X compact.

< 00 or dim Y < 00. Show that T

E L(X, Y) is

6.5 Bilinear Forms

137

The significance of compact operators T E L(X, X) lies in the fact that the equation Tx - AX = y (with X, y EX, Y given, X sought) has properties analogous to the finite-dimensional case. Theorem 6.4.12. (Riesz-Schauder theory) Let X be a Banach spacej let T E L(X, X) be compact. (a) For each A E ClJ\{O} one of the following alternatives holds:

(i) (T - AI)-l E L(X, X)

or

(ii) A is an eigenvalue.

In case (i) the equation Tx - AX = Y has a unique solution X E X for all Y E X. In case (ii) there exists a finite-dimensional eigenspace E(A, T) := kernel (T - AI) # {O}. All X E E(A,T)\{O} solve the eigenvalue problem Tx=Ax, x#O. (b) The spectrum q(T) ofT consists by definition of all eigenvalues and, if not T-l E L(X, X), A = O. There exist at most countably many eigenvalues which can only accumulate at zero. A E a(T) if and only if'). E a(T'). Furthermore,

dim(E(A, T» = dim(E(X, T'» < 00. (c) For A E q(T)\{O}, Tx - Ax = y has at least one solution x only if (y, X'}XXXI = 0 for all x' E E(5., T').

E

X if and

In Lemma 6.5.18 we need Lemma 6.4.13. Let Xc Y c Z be continuously embedded Banach spaces and let X C Y be compactly embedded. Then for every f > 0 there exists a GE such that (6.4.2) IIxlly ~ fllxllx + GEllxllz for all x E X. PROOF. Let f > 0 be fixed. The negation of (2) reads: There exists Xi E X with (lixilly - flixillx)/lIxiliz -+ 00. For Yi := (fllxillx)-lxi E X we thus have (IlYilly -1)/IlYillz -+ 00. From this one infers IIYillz -+ 0 and IIYilly > 1 for sufficiently large i. Since IIYilix ~ 1/f. and Xc Y is a compact embedding, a subsequence Yik converges to y* E Y. Now, IIYilly > 1 implies IIY*lIy ~ 1, i.e., y* # O. On the other hand, Yik also converges in Z to y* since Y c Z is continuously embedded. IIYiliz -+ 0 gives the contradiction sought: y* = o.

•

6.5 Bilinear Forms In the following let us assume that V is a Hilbert space. The mapping a(., .): V x V -+ IR is called a bilinear form if a(x, y + AZ)

= a(x, y) + Aa(x, z),

a(x + AY, z)

= a(x, z) + Aa(y, z)

138

6 Tools from Functional Analysis

for (x, y, Z E V, A E IR) (in the complex case one speaks of sesquilinear forms: a(x,Ay) = ~a(x,y)). a(·,·) is said to be continuous (or bounded) ifthere exists a C s such that la(x, y)l ~ Csllxllvllyllv

for all x, y E V.

(6.5.1)

Lemma 6.5.1. (a) To a continuous bilinear form one can assign a unique operator A E L(V, V') such that a(x, y)

= (Ax, y)v'xv

for all x, y E V,

IIAllvl J(x) for all z

J(x)

2f(y)

+ CEliz - xII},

•

=f x.

The term "V-elliptic" seems to indicate that to elliptic boundary value problems correspond V -elliptic bilinear forms. In general this is not the case. Rather, V-coercive forms will be assigned to the elliptic boundary value problems. Their definition necessitates the introduction of a Gelfand triple (cf. (6.3.7)): V cUe V'

(U = U', V C U continuous and densely embedded).

Definition 6.5.13. Let V cUe V' be a Gelfand triple. A bilinear form a(·, .) is said to be V-coerci ve if it is continuous and if there exists CK E m. and CE > 0 such that

(6.5.10)

142

6 Tools from Functional Analysis

Exercise 6.5.14. Set a(x,y):= a(x,y) +CK(x,Y)u with C K from (10). Let I: V --+ V' be the inclusion. Show that (a) the coercivity condition (10) is equivalent to the V -ellipticity of a. (b) If A E L(V, V') is associated to a(·, .), then so is A := A + C,.I to a{·, .). Why does A E L(V, V') hold?

The results of Riesz-Schauder theory (Theorem 6.4.12) transfer to A as soon as the embedding V C U is not only continuous but also compact. Theorem 6.5.15. Let V cUe V' be a Gelfand triple with compact embedding V cU. Let the bilinear form a{., .) be V -coercive with corresponding operator A. Let I: V --+ V' be the inclusion. (a) For each AE CU one of the following alternatives holds:

(A - AI)-l E L{V' , V) (ii) A is an eigenvalue. (i)

and (A' - Xl)-l E L{V', V),

In case (i) Ax - Ax = f and A'x· - 'Xx. = f are uniquely solvable for all f E V' (i.e., a(x, y) - A(X, y)u = fey) and a·(x·,y) -X(x., y)u = fey) for all y E V). In case (li) there exist finite-dimensional eigenspaces {O} =f E{A) := kernel (A - AI) and {O} =f E'(A) := kernel (A' - XI) such that Ax = Ax A'x·

= X*

= A{X, y)u

for x E E{A),

i.e.,

a{x, y)

for x* E E'(X),

i.e.,

a·(x*,y)

= X(x*,y)u

for all y E V, (6.5.11a) for all y E V. (6.5.11b)

(b) The spectrum u(A) of A consists of at most countably many eigenvalues which cannot accumulate in CU. A E u(A) if and only if X E U(A'). Pu.rthermore dimE(A) = dimE'(X) < 00. (c) For A E u{A), Ax - Ax = f E V' has at least one solution x E V if and only f..L E'{A), i.e., (f,x*}v'xv = (f,x*)u = 0 for all x· E E'{A).

PROOF. With V c U, V C V'is also a compact embedding, i.e., the inclusion I: V --+ V' is compact. A + CKI with CK from (10) satisfies A + CKI E L(V, V'), (A + CKI)-l E L(V', V) (cf. Exercise 14 a). Lemma 6.5.4 shows that K := (A + CK 1)-1 I: V --+ V is compact. Hence the RieszSchauder theory is applicable to K - 1'1. Since K -1'1= -1'(1 -1'-lK)

= -1'(A+CKI)-l{A+CKI -1'-lI} = -I'(A + CKI)-l(A - AI)

with A = 1/1' - CK, the statements of Theorem 6.4.12 transfer via K - 1'1 to A - AI = -1'-1 (A + CKI) (K - 1'1). •

6.5 Bilinear Forms

143

Remark 6.5.16. The spectrum u(A) has measure zero so that under the conditions of Theorem 15 the solvability of Ax - AX = f is guaranteed for almost all A. Problem (7) is solvable if not "accidentally" 0 E u(A). Lemma 6.5.17. Under the conditions of Theorem 15 the inequalities (4a,b) are equivalent.

PROOF. (4a) proves that A is injective, Le., 0 f/. u(A). Theorem 15a shows A-l E L(V', V) so that (4b) follows from Lemma 3. • Evidently a(·, .) remains V-coercive if one adds a multiple of (., ·)u. Generally, there holds

Lemma 6.5.18. Let a(·,·) be V-coercive where V cUe V'. Then a(·,·) + b(., .) is also V -coercive if the bilinear form b(., .) satisfies one of the following conditions: (a) for every € > 0 exists Ce such that (6.5.12a)

(b) Let the embeddings V c X and V C Y be continuous, with at least one of them compact. Let the following hold: Ib(x,x)1 ~ CBllxllxllxlly

for all X E V.

(6.5.12b)

(c) Let the embeddings V eX, V c Y be continuous. Let (12b) hold. For II . IIx or II . lIy assume that for every f. > 0 there exists a C~ such that

IIxlix ~ €lIxliv + C~llxliu

or

IIxlly ~

€lIxllv + C~lIxllu

for x E V. (6.5.12c)

= CE/2 with CE from (10). Then a(·,·) +b(.,.) satisfies the V-coercivity condition with C E /2 > 0 and C K + Ce instead of C E and CK. (b) Lemma 6.4.13 proves (12c). (c) Let the first inequality from (12c) hold, for example. Since the embedding V c Y is continuous, Cy exists with IIxllY ~ Cyllxllv. Choose €' = €/(2CB Cy) in (12c): PROOF. (a) Select €

Ib(x,x)1

~ CB(€'lIxliv +C~,"xllu)CYllxllv ~ ~lIxll~ +Kllxllvllxllu

with K = CBCyC~,. Since Kllxllvllxllu ~ ~"xll~ (5.3.10)), (12a) follows.

+ !K2€-1I1xll& (cf. •

7 Variational Formulation

7.1 Historical Remarks In the preceding chapters it was not possible to establish even for the Dirichlet problem of the potential equation (2.1.1a,b) whether, or under what conditions, a classical solution 1.£ E C 2 (fl) n CO(fl) exists. Green [I] took the view that his Green's function, described in 1828, always exists and that it provides the solution explicitly. This is not the case. Lebesgue proved in 1913 that for certain domains the Green function does not exist. Thomson (1847), Kelvin (1847), and Dirichlet offered a different line of reasoning. The Dirichlet integral 1(1.£):=

f

Ja

IVu(x)1 2 dx =

f tu~.(x)dx

Ja i=1

(7.1.1)

describes the energy in physics. With boundary values 1.£ = II' on r given, one seeks to minimise 1(1.£). This variational problem is equivalent to

1(1.£, v):=

fa

(Vu(x), Vv(x»dx = 0 for all v with v

= 0 on r.

(7.1.2)

The proof of the equivalence results from 1(1.£+ v) = 1(u)+21(u, v)+l(v) and l(v) ~ 0 for all v (cf. Theorem 6.5.12). Green's formula (2.2.5a) provides 1(1.£, v) = fa v.::1udx = 0 for all v with v = 0 on r such that .::1u = 0 follows. Thus, like (2), the variational problem 1(1.£) = min is equivalent to the Dirichlet problem .::1u = 0 in n, 1.£ = II' on r. The so-called Dirichlet principle states that l(u), since it is bounded from below by l(u) 2 0, must take a minimum for some u. According to the above considerations this would ensure the existence of a solution of the Dirichlet problem. In 1870, WeierstraB argued against this line of reasoning, stating that while there may exist an infimum of l(u) over {u E C2(fl) n CO(fl):u = II' on r} it need not necessarily be in this set. For example, the integral J(u):= fo1u 2(x)dx in {u E CO ([0, 1]):1.£(0) = 0,1.£(1) = I} never takes the value inf J(u) = O. Further, the following example due to Hadamard shows that no finite infimum of the Dirichlet integral need exist. Let r and II' be the polar coordinates in the circle fl = Kl(O). The function u(r, 11') = :E~=1 r n1 n- 2sin(n! O.

(7.2.10)

PROOF. Since lal = 1.81 = lone can identify a and.8 according to Da = 8/8xi, D~ = 8/8xj with indices i,j E {I,,,,, n}. For fixed x E n use (3) with ~ = Vu(x):

L lal,I~I=l

n

aa~(x)~a+~ =

L

aij(x)~i~j ;::: fl~12 = fIVu(x)12.

i,j=l

Integration over n yields a(u, u) ;::: fIn IVul 2 dx. Since In IVul 2 dx ;::: Gnluli (cf. Lemma 6.2.11), (10) follows with f' = fGn. • Corollary 7.2.4. The condition "n bounded" may be dropped if for a .8 = 0 one assumes aoo(x) ;::: 1J > 0 (instead of aoo = 0). Example 7.2.5. The Helmholtz equation -Llu the bilinear form

+u

=

=

f in n leads to

148

7 Variational Formulation

a(u, v):=

1rf: a

uz.(x)vz.(x) + u(x)v(x)] dx =

i=1

1 a

[(Vu, Vv)

+ uv] dx.

a(u,v) is the scalar product in HJ(D) (and H1(D)). The fact that a(u,u) = lul~ proves the HJ(n)-ellipticity.

Exercise 7.2.6. Let the assumptions of Theorem 3 or Corollary 4 be satisfied, except for the fact that the coefficients aaO and 0.0{3 (Ial = IPI = 1) of the first derivatives are arbitrary constants. Show that inequality (10) holds unchanged. Theorem 3 cannot easily be extended to the case m

> 1.

Theorem 7.2.7. Let the coefficients oj the principal part be constants: aa{3 = const Jor lal = IPI = m. FUrthermore, assume that aa{3 = 0 Jor 0 < lal + IPI :$ 2m - 1, aoo ~ 0 Jor a = P = O. Let L be uniJormly elliptic (cJ. (3)). F'u.rther let either n be bounded or aoo ~

'1 > O.

Then a(·,·) is Hlf(D)-elliptic.

PROOF. We continue u E Hlf(n) through '1.£ = 0 onto JRn • Theorem 6.2.21, Exercise 6.2.22, and inequality (3) show that a(u, '1.£)

-1

aoou2 dx =

a

L

1

L

aa{3

aaf3DauIJf3udx

lal,I{3I=m a lal,I{3I=m =

L

J.

D a uD{3udx

R"

aaf3

L)(ie )au (e)][(i e ){3u(e)]d{

lal,If3I=m

=

J. [ L

aa{3ea+{3l1u(e) 12 d{

R" lal,If3I=m

~ E r leI 2m lu(e)1 2 d{. lIt" Let 0.00 ~ '1 > O. There exists an E' > 0, so that Elel 2m ~ E' E~al::;m lea l2 - '1 for all e E JRn . From this follows EJ leI 2m lu(e)1 2d{ ~ E'lul m - '11'1.£16 and a(u,u) ~ E'lul~ (cf. Lemma 6.2.23). If n is bounded, use Lemma 6.2.11. • Having shown the Hlf(n)-ellipticity of the form a(·,·), we are now able to apply the general proofs after Theorem 6.5.9.

Theorem 7.2.8.

(Existence and uniqueness of weak solutions)!J a(·,·) is H{f(n)-elliptic then there exists a solution u E Hlf(D) oj Problem (9) which satisfies 1

lul m:$ CE IJI-m

(CE from (6.5.5)).

(7.2.11)

7.2 Equations with Homogeneous Dirichlet Boundary Conditions

Since (11) holds for all f E H-m(a) and u is equivalent to

= L-1 f

149

(cf. (9')), inequality (11)

(7.2.11')

The term "variational problem" for (9) goes back to the following statement, inferred from Theorem 6.5.12:

Theorem 7.2.9. Let a(·,·) be an H{f(a)-elliptic and symmetric bilinear form. Then (9) is equivalent to the variational problem find where

u E HQ'(a),

such that

J(u)::; J(v)

for all v E HQ'(a), (7.2.12a)

1 J(v) := 2a(v, v) - f(v).

(7.2.12b)

Attention. If a(·,·) is either not Ho(a)-elliptic or not symmetric, Problem (9) remains meaningful although the solution does not minimize the functional J(u). Example 7.2.10. The Poisson equation -Llu = f in a, u = 0 on leads to a(u,v) = In{Vu(x), Vv (x» dx. For a bounded domain a, a(.,.) is HJ(n)-elliptic (cf. Theorem 3), so that for any f E H-1(a) there exists exactly one (weak) solution u E HJ(a) of the Poisson equation. This is also the solution of the variational problem! In IVul 2 dx - feu) = min.

r,

A weaker condition than Ho(a)-ellipticity is the Ho(a)-coercivity: a(u,u) ~ €Iul~ - Glula.

= 1, and let the coefficients a"'{3 E LOO(a) satisfy condition (3) of uniform ellipticity. Then a(·,·) is HJ(a)-coercive.

Theorem 7.2.11. Let m

PROOF. We write L as L = LI + L Il , with LI satisfying the conditions of Theorem 3, resp. Corollary 4 if a is not bounded, and LII containing only • derivatives of order ::; 1. Then we can apply the following lemma.

Lemma 7.2.12. Let a(·, .) = a' (-, .) + a" (-, .) be decomposed such that a' (', .) is HQ(ll)-elliptic, or perhaps only H{f(ll)-coercive, while a"(u,u)

=

L lal.I.BI$m lal+I.BI.(e, V)L~(a)

for all v E HO'(n)

(7.2.13)

has countably many eigenvalues which do not accumulate in CIJ.

PROOF. Since for bounded n the embedding V:= H{j(n) c U := L2(n) is compact (cf. Theorem 604.8b), Theorem 6.5.15 is applicable. •

7.3 Inhomogeneous Dirichlet Boundary Conditions Next, we consider the boundary value problem Lu = g in

n,

11.

= 0 there exists an i such that d(u, Vi) ~ E. From the assumption (4b) there is, for each '1.£ E V and E > 0, awE Vi with 11'1.£ - wllv ~ E. Therefore we have w E Vi for an i E IN. That d(u, Vi) ~ 11'1.£ - wllv ~ E then proves (4a). The convergence u i ---+ '1.£ implies the uniqueness of the solution u. (b) The next thing to show is that the image W := {Lv:v E V} C V' of the operator L: V ---+ V' associated to a(·, .) is closed. For each fEW there is a '1.£ E V with Lu = f, so that part (a) of this proof suffices to show the convergence u i ---+ u. Since lIuiliv ~ IIfllv,ll, it follows that lIuliv = lim lIuiliv ~ IIfllv' /€. Let f" E W be a sequence with f" ---+ in V' and f" = Lu". The Cauchy convergence IIf" - f#lIv' ---+ 0 shows 11'1.£" -u#lIv ~ IIf" - f#lIv' I€'---+ 0 so that the limit u· = limu" E V exists. The continuity of L E L(V, V') shows that = lim f" = lim Lu" = Lu*, and thus that E W. Hence W is closed. (c) In order to demonstrate the existence of a solution to the problem (1.1) we have to show the surjectivity of L: V ---+ V'. If L were not surjective (Le., W =j:. V'), there would be an f E W.L with Ilfllv' = 1. Let J v :V ---+ V' be the Riesz isomorphism (cf. Corollary 6.3.7). Set v := -JV1 f E V. It follows that

U

r

r

r

f(v)

= {j,v}v'xv = -(f,J)v' = -1,

a(u,v)

= (Lu,v}v'xv = 0

for all '1.£ E V. Let u i E Vi be the llitz-Galerkin solutions. They must also satisfy a(ui , v) = 0, Le., a(ui , v) - f(v) = 1. We split up v into vi + wi, where vi E Vi, By (4a), with v in place of '1.£, one may guarantee that IIwiliv ---+ O. This shows 1

= a(ui,v) -

f(v)

= a(ui , vi) -

= a(ui,wi ) -

f(v i ) + a(ui,wi ) - f(w i ) f(w i )

and 1 = la(ui,v) - f(v)1 ~ [Cslluillv

+ Ilfllv']

Ilwiliv.

Since lIuiliv ~ IIfllv'/€ is uniformly bounded and Ilwiliv ---+ 0 this amounts to a contradiction. Therefore L must be surjective, so that for each f E V' there exists a solution '1.£ to the equation Lu = f, i.e., to the problem (1.1). •

170

8 The Method of Finite Elements

Corollary 8.2.3. The requirement (1.12) with eN. ;::: l > 0 from Theorem 2 is satisfied with l:= CE i/ a(.,.) is V-elliptic: a(u, u) ;::: CEllull~.

Exercise 8.2.4. Show that (4a) implies that U:l Vi is dense in V. Let QN be the orthogonal projection onto VN (cf. Exercise 8.1.6c). The factors L:V - V', QN:V' - Vlv = VN, L1/:Vlv = VN - VN C V can be composed to give (8.2.5a)

Exercise 8.2.5. Show there is also the representation SN =PL-1p*L.

(8.2.5a')

Lemma 8.2.6. SN is the projection onto VN and is called the Ritz projection. It sends the solutionu o/the problem (1.1) to the Ritz-Galerkin solution uN E VN: uN = SNU. Assuming (1.2) and (1.12) we have IISNllv 0, is difficult to prove, except for V-elliptic bilinear forms. However, the following theorem shows that this condition does hold for subspaces approximating well enough. Theorem 8.2.8. Let the bilinear form a(-, .) be V -coercive, where V cUe V'is a continuous, dense, and compact embedding. Let Problem (1.1) be solvable for all f E v'. Assume that (4a) holds for the subspaces V. C V. For larye enough i the stability condition (1.12) is then satisfied with EN. 2 E > o.

The proof of this will be postponed to a supplement to Lemma 11.2.7.

8.3 Finite Elements 8.3.1 Introduction: Linear Elements for

n = (a, b)

A13 soon as the dimension N = dim VN becomes larger the essential disadvantage of the general Ritz-Galerkin method becomes apparent. The matrix L is in general full, i.e., Lij i- 0 for all i,j = 1, ... , N. Therefore one needs N 2 integrations to obtain the values of Lij = a( bj , bi ) = I a ... , whether exactly or approximately. The final solution of the system of equations Lu = f requires O(N3) operations. A13 soon as N is no longer small the general Ritz-Galerkin method therefore turns out to be unusable. A glance at the difference method shows that the matrices Lh which occur there are sparse. Thus it is natural to wonder if it is possible to choose the basis {bi> ... , bN } so that the stiffness matrix Lij = a(bj , bi) is also sparse. The best situation would be that the bi were orthogonal with respect to a(·, .): a(bj,bi ) for i i-j. However, such a basis can be found only for special model problems such as the one in Example 8.1.11. Instead we shall base our further considerations on Remark 8.3.1. Let the bilinear form a(·,·) be given by (7.2.6). Let Bi be the interior of the support supp(b;.) of the basis function bi, i.e., Bi := supp(bi )\8supp(bi ). A sufficient condition that ensures Lij = a(bj , bi) = 0 is BinBj = 0.

PROOF. The integration a(bj,bi )

= Ia ... can be restricted to Bi n B j .

•

In order to be able to apply Remark 1 the basis functions should have as small supports as possible. In constructing them in general one goes about

172

8 The Method of Finite Elements

n

it from the desired goal: one defines partitions of into small pieces, the socalled finite elements, from which the supports of hi are pieced together. Ai; an introduction let us investigate the one-dimensional boundary value problem

-u"(X)

= g(x)

for a < x < b,

u(a)

= u(b) = O.

(8.3.1)

ABsume we have a partition of the interval [a, b] given by a = Xo < Xl < ... < XN+1 = h. Denote the interval pieces by Ii := (Xi-l,Xi) (1 ~ i ~ N + 1). For the subspace VN C HJ(a, b) let us choose the piecewise linear functions:

VN ={u E CO([a, b]): the restriction of u to Ii (1 ~ i ~ N is linear; u(a) = u(b) = O}

+ 1)

(8.3.2)

See the figure below.

X

Xs =b Figure 8.S.1. (a) a piecewise linear function, (b) a basis function

The continuity "u E CO([a, b])" is equivalent to continuity at the nodes Xi (1 ~ i ~ N) : U(Xi + 0) = U(Xi - 0).

Remark 8.3.2. (1 ~ i ~ N): u(x)

U

E VN is uniquely determined by its node values U(Xi)

= [U(Xi)(XHl -

x) + u(xHd(x - Xi)]/[XHl - Xi] for X E I i +1, (8.3.3a)

where u(xo) = U(XN+l) = O. From Theorem 6.2.42 we conclude that VN C HJ(a,b). The (weak) derivative u' E L2(a,b) is piecewise constant: (8.3.3b) The basis functions can be defined (see Figure 1b) by

bi(x)

=

(x - xi-d/(Xi - xi-d { (Xi+1 - X)/(Xi+1 - Xi)

o

for Xi-l < X ~ Xi for Xi < X < Xi+1, otherwise.

1~ i ~ N

(8.3.4)

8.3 Finite Elements

173

Remark 2 then has as a consequence

Remark 8.3.3. One has the representation u = ~!1 u(xi)bi . The supports of the functions bi are 1i - 1 u1i = [Xi-l,Xi+1]' The Bi from Remark 1 is now Bi = (Xi-1,XHt)·

The weak formulation of the boundary value problem (1) is a(u, v) = f(v) with a(u,v):=

lb

f(v):=

u'v' dx,

lb

gvdx

for u,v E HJ(a, b).

By Remark 1 we have, for the matrix elements, Lij 2, since then B. n Bj = 0. For Ii - jl ~ lone obtains L.,.-1 = a(bi _l, bi ) = Lii

= a(bi , bi ) = 1

al•

1al.-lal•

Xi - Xi-1 Xi - Xi-1

(Xi -

Xi_t}-2

-1

1

dx

+

Xi)-2 dx (8.3.5a)

:.;.

= 1/(Xi - Xi-1)

=

dx = -1/(xi - xi-d,

1al.+l (XH1 -

2:._1

L i ,H1

= 0 as soon as Ii - j I ~

+ 1/(XH1 -

Xi),

-1/(XH1 - Xi).

The right-hand side f

1al

= (ft, ... , fN)T is given by (5b):

.+ 1

fi = f(b.) =

gbi dx

%'-1

=

1 Xi - Xi-1

1 alO g(x)(x al.-l

xi-d dx +

1 al.+

1 Xi+l - Xi ""

1

g(X)(XH1 - x) dx.

(8.3.5b)

Remark 8.3.4. The system of equations Lu = f, given in (5a,b), is tridiagonal, thus, in particular, sparse. For an equidistant partitioning Xi := a + ih with h (b - a)/(N + 1) the coefficients are Li,i±1 = -1/h, Li,i = 2/h, fi = h fo [g(Xi + th) + g(Xi - th)](1 - t) dt. If 9 E COCa, b), then by the intermediate value theorem fi = hg(Xi + eih), with leil < 1. Therefore the system of equations Lu = f is, up to a scaling, identical with the difference equation LhUh = !h in Section 4.1, if one defines fh by fh(Xi) := Ii = g(Xi + eih) instead of by fh(Xi) = g(Xi): then L = hLh , f = hfh.

::=

This example shows that it is possible to find basis functions with small supports so that L is relatively easy to calculate. In addition, the similarity is apparent between the resulting discretisation method and difference methods. Finally consider the numerical evaluation of the integral (5b).

Exercise 8.3.5. Show that the Gaussian quadrature formula for with bi(x) as weight function and one support point is:

f:::

1 1

gbi dx

174

8 The Method of Finite Elements

. '= Xi+l f,•. 2

Xi-l

9

(Xi-l

+ X.3 + XH1) .

(8.3.5c)

What is the formula for a partition of equal intervals? If one replaces the Dirichlet boundary conditions in (1) by the Neumann condition then V = Hl(a,b). The subspace VN C Hl(a,b) results from (2) after removal of the condition u(a) = u(b) = O. In order that dim VN = N the numbering has to be changed: The partition of (a,b) is given by a = Xl < X2

< ... and C from part (a). Thus we • have assertion (2) with C:= l/CE •

°

° °

°

Lemma 8.4.2. Let Th := hT := {(x,y): x,y ~ 0, x '1.£ E H2(Th) there holds with 1f31 ~ 2

+y

~

h}. For each

IID13ullh(Th) ~ C{h2-2113I[u2(0, 0) + '1.£2(0, h) + u2(h, 0)1

+ h4-21131

L

(8.4.3)

II DOIU Il12(Th)}'

1011=2 with C independent of '1.£, f3, and h.

PROOF. Let v(~, 11) := u(~h, 11h) E H2(T). The derivatives with respect to (x,y) and to (~,11) = (x,y)/h are related by D~.II = h- I13I D:.". For each 1f31 ~ 2 Lemma 1 gives II D13u Il12(Th) =

JLID~.lIuI2

dxdy =

J£

Ih-I13ID:'"vI2h2

~ d11

= h2-2113II1D13vIl12(T) ~ h2-2113ll1vllk2(T)

~ Ch2-2113I[V2(0, 0) + v 2(1,0)

+ v2(0, 1) +

L

IIDOIvI112(T)j·

1011=2 Since v(O, 0) = '1.£(0,0), v(l, 0) = u(h, 0), v(O, 1) = '1.£(0, h), and IIDl"vIl12(T) = h2I1D~.flUIl12(Th)' one may conclude assertion (3). •

8.4 Error Estimates for Finite Element Methods

187

Lemma 8.4.3. Let T be an arbitrary triangle with Side lengths $ h max ,

Interior angles

~

ao

> O.

(8.4.4)

For each u E H2(T) and 1,81 $ 2, there holds

IID.6 u lli2(T) $ C(ao)

[h~~I.61 L

lu(x)1 2 +

x vel1ex of T

h!t~I.61 lal=2 L IIDaUlli2(T)] , (8.4.5)

where C(ao) depends only on ao and not on u, ,8, or h.

PROOF. Let h $ h max be one of the lengths of the sides of T. In a way similar to that in Exercise 8.3.14a, let cp: Th -+ T be the linear transformation that maps Th onto T. Then V(€,71) := U(CP(€,71)) belongs to H2(Th). Under the conditon (4) we know that the determinant Idetcp'l E [l/K(ao),K(ao)] is bounded both above and below. From .6 2 "" .....8' 2 II D a:,yulI£2(T) $ C1(ao) L.J IIq,f/ v IIL2(Th)' 1.6'1=1.61 (3), and

L

L

II Dl,f/vII12(Th) $ C2 (ao) IID~,yulli2(T)' lal=2 lal=2 there then follows (5). From Lemma 3 follows the important result:

•

Theorem 8.4.4. Assume that conditions (la-c) hold for r, VN, and V. Let ao be the smallest interior angle of all Ti E r, while h is the maximum length of the sides of all 11 E r. Then inf lIu-vIlHlc(O) $ C'(ao)h2- lc lluIlH2(0) for k = 0, 1 and all u E H2(il)n V. lIEVN

(8.4.6) PROOF. For a u E H2(il) [resp. u E H2(il) n HJ(il), if V = HJ(il)] one chooses v := l:i u(xi)bi E VN, i.e., v E VN with v(Xi) = U(Xi) at the [inner] nodes Xi. Since w := u - v vanishes at the vertices of each 11 E r and Da w = nau for lal = 2 (because Dav = 0 for all v E VN), inequality (5) implies the estimate

IID.6 w Il12(0)

=

L

IID.B w Il1 2 (T,)

L [C(ao)h4-21J31 L II Dau Il12(T,)1 lal=2 = h4-21J3IC(ao) L II Dau Il1 $

T,Er

2 (Q)

lal=2 $ h4-21.6IC(ao)lIull~2(0).

188

8 The Method of Finite Elements

Summation over 1.81 :::;; k now proves the inequality (6).

•

Let hT be the longest side of T E T, then h := max{hT:T E T} is the longest side-length which occurs in T. The parameter h is the essential one and will be used as an index in the sequel: T = Th. Denote by PT the radius of the inscribed circle in T E T. The ratio hTI PT tends to infinity exactly when the smallest interior angle tends to zero. If one has the bound max {~I PT: T E T} :::;; const for a family of triangulations, then one calls the family quasi-uniform. In this condition the constant should be the same for all T in the family {Th}. A family {Th} is called uniform if it fulfils the stronger requirement hi min{PT: T E T} :::;; const. A family {Th} which is both unform and for which h -+ 0 is called regular by Ciarlet [1, §3.1]. In the construction of triangulations it should be noted that with increasing refinement (h" -+ 0 ) the interior angle may not also be reduced. Strategies for the systematic construction of quasi-uniform sequences Vh" can be found, for example, in Ha.ckbusch [1, §3.8.2]. The case k = 1 of Theorem 4 may now be written as follows:

Theorem 8.4.5. Assume conditions (1a-c) hold for a sequence of quasiuniform triangulations T". Then there exists a constant C, such that for all h = h" and Vh = Vh" there holds the following estimate: inf 1J€V"

lu - vb : :; Chlul2

for all

u E H2(n) n V.

(8.4.6')

The combination of this theorem with Theorem 8.2.1 gives

Theorem 8.4.6. Assume conditions (1a-c) hold for a sequence of quasiuniform triangulations T". Let the bilinear form fulfil conditions (1.2) and (1.12). Suppose the constants fN := fh" > 0 from (1.12) are bounded below by fh" ~ E > 0 (cf. Corollary 8.2.3). Let Problem (1.1) have the solution u E H2(n) n V. Let u h E Vh be the finite-element solution. Then there exists a constant C, which does not depend on u, h = h" or 11, such that (8.4.7) The combination of the Theorems 5 and 8.2.2 can be written:

Theorem 8.4.7. Assume conditions (la-c) hold for a sequence of quasiuniform triangulations T" with h" -+ O. Let the bilinear form fulfil (1.2) and (1.12) with fh" ~ l > O. Then the problem (1.1) has a unique solution u E V, and the finite-element solution u h " E Vh " converges to u: lu-uh "ll-+0

(11-+00).

(8.4.8)

PROOF. Let u E V and f > 0 be arbitrary. Since H2(n) n V is dense in V, there exists U e E H2(n) n V with lu - ueb :::;; f/2. From (6) and h" -+ 0 we

8.4 Error Estimates for Finite Element Methods

189

see there are v and VE e Vh " with IUE - VEil :$ f./2; thus lu - vEil :$ f.. This proves (2.4a). • Theorem 7 proves convergence without restrictive conditions on u. On the other hand the order of convergence O( h) in the estimate (7) requires the assumption that the solution u e V lies in H2(n). AB will later become clear, this assumption is hardly to be taken as fulfilled in every situation. A weaker assumption is u e V n HB(n) with s e (1,2). The corresponding result is as follows.

Theorem 8.4.8. Assume that in the assumptions of Theorem 6 u e V n H2(Q) is replaced by u e V n HB(Q) with 1 :$ s :$ 2. Let n be sufficiently smooth. Then there holds (8.4.9) The proof uses a generalisation of Theorem 4:

Lemma 8.4.9. Under the assumptions of Theorem 4 and suitable conditions on n, there holds inf lu - vh :$ C"(ao)h B- l lul s for s e [1,2], u e HS(n) n V. (8.4.10) VEVh

PROOF. The proof is based on an interpolation argument that will not be further explained here. Equation (10) holds for s = 2 (cf. (7» and for s = 1, since influ - vI! :$ lu - Olt = lull' From this follows (10) for s e (1,2) with the norm I· Is of the interpolating space [Hl(n) n V, H2(n) n V]s-l (cf. Lions-Magenes [1]), which under suitable assumptions on n coincides with HS(n)nV. • For s = 1 the right-hand side of (10) becomes const ·Iult. In fact the estimate inf{lu-vll:v e Vh } :$ lull is the best possible. To prove this ,choose u .1 Vh (orthogonal with respect to I·h). On the other side s = 2 is the maximal value for which the estimates (9) and (10) can hold. Even u e c=(n) permits no better order of approximation than O(h)! The Ritz projections SN : V ~ VN introduced in (2.5a) will now be written Sh : V ~ Vh. The inequality (9) becomes lu - Shull/luis :$ Ch s - l , and this proves

Corollary 8.4.10. Assume conditions (la-c) forT, Vh and V. Let the bilinear form fulfil (1.2) and (1.12). The Ritz projection Sh satisfies the estimate

III -

Sh IIHI (J2)+-H2(J2)nv :$ Chl.

(8.4.11)

Under the asumptions of Lemma 9 there holds

III -

ShIlHl(.a)+-HS(J2)nV :$ Ch s-

1

for all 1 :$ s :$ 2.

(8.4.11')

190

8 The Method of Finite Elements

8.4.2 L2 and H8 Estimates for Linear Elements

According to Theorem 6 O(h), is the optimal order of convergence. This result seems to contradict the O(h2) convergence of the five-point formula (cf. Section 4.5), since the finite-element method with a particular triangulation is almost identical to the five-point formula (cf. Exercise 8.3.13). The reason for this is that the error u - u h is measured in the I . It norm. The estimate inf{lu -vlo: v E Vh} ::;; Gh21ul2 in Theorem 4 suggests the conjecture that the I ·10 norm of the error is of the order O(h2): lu - uhlo ::;; Gh21u12. However, this statement is false without further assumptions. Up to this point only the existence of a weak solution u E V (e.g., V = HJ(.o) or V = H1(.o» has been guaranteed. But in Theorem 6 we needed the stronger assumption u E H2(.o) n V. A similar regularity condition will also be imposed on the adjoint problem to (1.1) Find u E V with a*(u, v)

= /(v)

for all v E V,

(8.4.12)

which uses the adjoint bilinear form a·(u, v) := a(v, u). For / E L2(il) C V', the value /(v) becomes (f,v)L2(O). The regularity condition is: For each / E L 2 (.o) the problem (12)

(8.4.13)

has a solution u E H2(il) n V with lul2 ::;; GRillo.

In Section 9 we shall see that this statement holds for sufficiently smooth domains .o. Theorem 8.4.11. (Aubin-Nitsche) Assume (13), (1.2), and (1.12) withf'.N = f'.h ~ E > 0, and (8.4.14) Let problem (1.1) have the solution u E V. Let u h E Vh C V be the finiteelement solution. Then, with a constant G1 independent of u and h, (8.4.15a)

If the solution u also belongs to H2(il) independent of u and h, such that

n V,

then there is a constant G2, (8.4.I5b)

Prom Theorem 4, it is sufficent to ensure (14) that Vh be the space of finite elements of an admissible and quasi-uniform triangulation. (cf. Nitsche [1]). For each e := u - u h as the solution of (12) for / = e:

PROOF

a(v,w)

= (e,vk,(o)

E

L2(.o) define w

for all v E V.

E

H2(il) n V (8.4.16a)

8.4 Error Estimates for Finite Element Methods

191

Corresponding to w there is, by (14), a w h E Vh with

Iw - whit ~ Cohlwl2 ~ CoCRhlelo. Equation (2.3) is a(e,v)

(8.4. 16b)

= 0 for all v E Vh; therefore, in particular, we have (8.4.16c)

From (16a-c) we obtain lel~

= (e, e)L2(S1) = a(e, w) = a(e, w - w h ) ~ Csleltlw - whh ~ CslehCoCRhlelo,

and thus (8.4.16d) From (2.1) one deduces

leh = lu -

uhlv ~ (1 + CS/€h) inf

vEV"

lu -

vlv ~ (1 + CS/f"}lulv

so that (15a) follows with C 1 := CsCoCR(l + Cs/'i). If u E H2(il), one may use the inequality (7), leh ~ Chluh, and deduce (15b) with C 2 := CSCOCRC .

•

Corollary 8.4.12. The inequalities (15a) and (15b) are equivalent to the properties (17a) and (17b) of the Ritz projection Sh:

III III -

Shll£2(S1) O. Assume problem (1.1) has a solution u E V n Hk+1(n). Then the finite-element solution u h E Vh satisfies the inequality (8.5.2) lu - uhh ~ Ch k lulk+l' The Ritz projection satisfies

III -

Shllv+-Hk+l(o)nv ~ Chk.

The estimate (4.9) now holds for s E [1,k] ifu E HB(n)nv. With suitable regularity conditions, there are, as in Theorem 8.4.11, the error estimates

lu - uhlo ~ Ch k+1lulk+l, lIu - u h IIHk-l (0)' ~ Ch2k lulk+1'

(8.5.3a) (8.5.3b)

194

8 The Method of Finite Elements

For (3b) one needs, for instance, Hk+l-regularity: For each f E Hk-1(D) the adjoint problem (4.12) has a solution u E H k +1(D). Remark 8.5.2. Under suitable assumptions the inequality (4.20) holds for 1- k 5 t 5 1 5 s 5 k + 1. For negative t the norm I· It should be understood as the dual norm of H-t(D).

8.5.2 Finite Elements for Equations of Higher Order 8.5.2.1 Introduction: The One-Dimensional Biharmonic Equation All the spaces of finite elements, Vh, constructed so far are useless for equations of order 2m > 2, since Vh ct. Hm(D). According to Example 6.2.5, in order to have Vh C H2(n) it is necessary that not only the function u but also its derivatives Ux, change continuously between elements. The ansatz functions must therefore be piecewise smooth and globally in C 1 (D). As a model problem we introduce the one-dimensional biharmonic equation

ul/I/(x) = g(x) for 0

< x < 1, u(O)

= u'(O)

= u(l) =

u' (I) = 0

which becomes in its weak formulation a(u,v)=f(v) forallvEHg(O,I), where

a(u, v)

=

11

ul/vl/ dx,

f(v)

= fa1 gvdx.

(8.5.4)

Divide the interval n = (0,1) into equal subintervals of length h. Piecewise linear functions (cf. Figure 8.3.1) can be viewed as linear spline functions, so that it is natural to define Vh as the space of cubic splines (with u = u' = 0 for x = 0 and x = 1) (cf. Stoer [I, §2.4) and Stoer-Bulirsch [1, §2.4)). We can take as basis functions B-splines, whose supports in general consist of four subintervals (cf. Figure 1). Since cubic spline functions belong to 02(0,1), they are not just in HJ(O, 1), but even in H3(0, 1) n HJ(O, 1) .

•

.~.

•

Figure 8.5.1. B-spline

Simpler yet than working with spline functions is to use cubic Hermi te interpolation: Vh := {u E C 1(0, l):u cubic on each subinterval,

u(O) = u'(O) = u(l) = u'(l) = O}.

(8.5.5)

8.5 Generalisations

195

To each of the inner nodes Xj = j h there are two basis functions bli and b2i with bH(Xi) = 1, bii(Xi) = 0, b2i(Xi) = 0, b2i (Xi) = 1, bki(Xj) = bki(Xj) = 0 for k = 1,2 (j # i) (cf. Figure 2) .

.6.

.~

(b)

(a)

..

Figure 8.5.2. Hermite basis functions

The support consist of only two subintervals. The expressions are ~ x ~ xi+1. 2 b2i (X) = (h -Ix - xil)2(x - xi)/h for Xi-1 ~ x ~ Xi+1. bji(x) = 0 for x ¢ [Xi-l,Xi+1]'

b1i (x) = (h -Ix - xil)2(2lx - xii

+ h)/h3 for Xi-1

Exercise 8.5.3. Let Uti and U2i (0 < i < l/h) be the coefficients of the expression u(x) = I:i[Utib1i(x) + U2i~i(X)] E Vh. Show that the coefficients of the finite-element solution of the problem (4) are given by the equations h-3[-12u1.i_l + 24uli - 12u1.i+l] + h- 2[-6u2,i_l + 6U2,i+d h-2[6ul,i_l - 6Ul.i+1] +

h- 1 [ 2u2.i_l

= i1£,

+ 8U2i + 2U2,i+l] = hi

(8.5.6)

The system of equations (6) differs completely from the difference equations (cf. Section 5.3.3), since in Equation (6) there appear values Uti of the functions at the nodes together with the values U2i of the derivatives at the nodes.

8.5.2.2 The Two-Dimensional Case The ansatz (5) can be carried over to [1 c if one starts with a partition into rectangular elements. The ansatz function is bicubic:

ne

3

Vh:= {U E 0 1([1): U = Un

= 0 on r,

U=

L

cx."p,x"y'"

",p,=o on each rectangle of the partition.}. At each inner node one may prescribe the four values u, Ua:, u lI ' ua:lI' Consequently there are four unknowns and four basis functions bli(X, y), ... ,b4i(X, y) belonging to each node. The latter are products bji(x)bki(y) (j, k = 1,2) of the one-dimensional basis functions described in Section 8.5.2.1 (cf. MeisMarcowitz [1, Figure 15.22], Schwarz [1, §2.6.1]). If one starts from a triangulation T, a fifth-order ansatz gives what is wanted: u(x) = I:1"1::;5 a"x" on 11 E T. The number of degrees of freedom is

196

8 The Method of Finite Elements

21 (the number of II E Z2 with II ~ 0, 1111:5 5). For the nodes one chooses the points Xl, ... ,x6 in Figure 8.3.Sa. At the vertices xl, x 2,:x3 one prescribes 6 values {Dl'u(xj): IILI :5 2}. The 3 remaining degrees offreedom result from giving the normal derivatives 8u(xi )/8n at the midpoints of the sides x4,x5,x6. Notice that if two neighbouring triangles (T and t in Figure 8.3.8a) have the same vertex values {Dl'u(xj): IILI :5 2, j = 1,2} and the same normal derivative at X4, then both '1.£ and Vu are continuous on the shared side, i.e., Vh C H2(f1). According to whether V = H6(n), V = H2(n) n HJ(n) or V = H2(f1) one has to set '1.£, or the first derivatives, to be zero at the boundary nodes Xi.

Remark 8.5.4. The finite-element space Vh C H2(f1) described here can of course be used for differential equations of the order 2m = 2. 8.5.2.3 Estimating Errors Instead of (5.1) one obtains

where k ~ m depends on the order of the polynomial ansatz (e.g., k = 3 for cubic splines, cubic [resp. bicubic] Hermite interpolation). Here m = 2 for the biharmonic equation. At; in Theorem 8.5.1, there follows from (7) the error estimate (8.5.8a) for the finite-element solutionu h E Vh . Under suitable regularity assumptions one gets (2m - k - 1 :5 t

:5 m :5 s :5 k + 1)

(8.5.8b)

(cf. Remark 8.5.2). The maximal order of convergence 2(k - m) + 2 results for s = k + 1, t = 2m - k - 1 and requires '1.£ E Hk+l(n) n V. In addition each solution of the adjoint problem (12) with f E Hk+1-2m(n) must belong to Hk+l(f1).

8.6 Finite Elements for Non-Polygonal Regions Since the union of triangles and parallelograms only generates polygonal regions a polygonal shape was assumed in (3.6). The finite-element method is, however, in no way restricted to just such regions. On the contrary finite elements can readily be adapted to curved boundaries.

8.6 Finite Elements for Non-Polygonal Regions

197

Figure 8.6.1. Gmvilinear boundaries

Let V = H1(il) and let il be arbitrary. The triangulation T can be so chosen that (3.7a,b,d) hold and the "outer" triangles have two vertices on r = 8il as in Figure 1. In the convex case of Figure 1a one extends the linear functions defined on Tonto T := TUB. In the case that the boundary is concave (see Figure lb) one should replace T by T := T\B. One copes correspondingly in the situation of Figure lc. The nodes and the expressions for the basis functions remain undisturbed by these changes. is the union of the closures of all the inner triangles 11 E T and all modified triangles T which lie on the boundary. All the properties and results of Sections 8.3-8.4 extend to the new situation. The only difficulty is of a practical sort: To calculate L and f one has to work out integrals over the triangle T which involves arcs. Assume now V = HJ (il). The previous construction will not be a success, since the extensions of the linear functions to T will not vanish on the boundary piece r n at. Thus Vh c HJ(il) will not be satisfied. As long as the region n is convex, only the situation shown in Figure 1a occurs, and u h may be extended to BeT by u h = O. In the case of Figure lb, one must set the values at the inner nodes to zero, so that u h = 0 on T = T\B, and in particular u h = 0 on n 8(T\B). All in all, what results is that the support of any u h E Vh is in a polygonal region inscribed in il. One interpretation is the following: In the boundary value problem, il should be replaced by an approximating region ilh C il (cf. Theorem 2.4.6). The finite-element solution described agrees with that which would result from the smaller region. However, the error estimate in Theorem 8.4.4 only holds for the smaller region ilh: inf1JEvh lIu - VIlHl(Oh) ::; ChlluIlH2(O)' Since v = 0 on il\ilh for any v E Vh one should also estimate lIuIlHl(OWh)' Now n\ilh is contained in a strip 8 6 = {x E ilh:dist(x, r) ::; o} of width 0 := max{dist(x, r): x E il\nh}. One can estimate as follows:

n

r

If n\nh consists only of arc segments B as in Figure la (for instance, in the case of a convex region), and if il E C1,l, then 0 = O(h2). From this follows the estimate inf1JEvh lIu - VIlHl(O) S Ch2I1uIlH2(O), as for a polygonal

198

8 The Method of Finite Elements

region. If, however, as in Figure Ib the whole triangle is part of fl\flh , then 8 becomes O(h) and the approximation worsens to inf1JEvh lIu - VIlHl(n) ::; Chl/2I1uIlH2(S1) (cf. Strang-Fix [1, p. 192]). In order to adapt the trianglar or parallelogram elements better to a curved boundary one can use the technique of isoparametric finite elements. From Figure 8.3.4 we see that the reference triangle T may be mapped into an arbitrary triangle f E T by an affine transformation cf> : (e,1]) 1-+ Xl + e(x2 - Xl) + !}(xS - Xl) . The linear [resp. in the case of §8.3.4, quadratic] function u(x) on T can be expressed as the image, vee, 1]) = u(cf>(e, 1]», of a linear [resp. quadratic] function von T. We now replace the affine transformation of triangles cf> by a more general quadratic mapping

cf>(e,1]) := (

al + ~e + as1] + a4e2 + a5e1] + ~1]2 ) : T _ a7 + ase + ag1] + alOe2 + alle1] + al21]2

f

C

JR2 .

The image f is a curvilinear triangle. The coefficients aI, ... , al2 are uniquely determined by the nodes Xl, ... , x 6 of f which are the images of the vertices and midpoints of the sides of the reference triangle T (cf. Figure 2). The triangulation T used so far can be replaced by a "triangulation" by triangles with curved sides, if neighbouring elements have the same arcs as common boundaries and the midpoints also coincide. The ansatz functions on f E f have the form u(x) = v(cf>-l(x», where v(e,1]) is linear [resp. quadratic] in and 1]. The resulting finite-element space is the space of isoparametric linear [resp. quadratic] elements (cf. Strang-Fix [1], Ciarlet [1], Schwarz [1], Zienkiewicz [1]).

e

Figure 8.6.2. Mapping of the reference triangle T to the curvilinear triangle T In general, there is no reason to use curvilinear triangles in the interior of n. AB in Figure 3, one chooses ordinary triangles in the interior (Le., the quadratic transformation cf> is again taken to be linear). A boundary triangle, like f in Figure 3, is, on the other hand, defined as follows: Xl and x 2 are the vertices of f which lie on r. One chooses another boundary point x4 between Xl and x 2 and requires that the boundary af of the triangle cuts the boundary r of the region at Xl, x4, and x 2 .

8.7 Additional Remarks

199

Figure 8.6.S. lsoparametric triangular dissection

Remark 8.6.1. Section 5.2.2 shows that, in the case of other than Dirichlet boundary conditions, the construction of difference schemes is increasingly complicated. Instead one may restrict the usual difference methods to the inner grid points, and near the boundary discretise by using (e.g., isoparametric) finite elements.

8.7 Additional Remarks It is not possible to consider here many of the details about, and modifications of, finite-element methods. In the following, individual cases are just treated summarily. B.7.1 Non-Conformal Elements

Condition (1.3), Vh c V, characterises conformal finite-element methods. V are called non-conformal. An example Discretisations for which Vh of a non-conformal element is the "Wilson rectangle". For this method one starts with a dissection into rectangles R;, = (Xli, X2i) X (YH, Y2i) and assumes the function to be quadratic on R;,:

ct

Vh := {u quadratic on R;" u continuous at vertices of the rectangle R;,}. Notice that u E Vh is required to be continuous only at the nodes, and not in the interior of D. A function which is not continuous along a side of R;, cannot belong to Hi(D): Vh V = Hi(D). A possible basis for Vh is the following. For each node x j , let bj be the basis function belonging to the bilinear elements (cr. Section 8.3.3). The quadratic ansatz chosen here has two

ct

200

8 The Method of Finite Elements

additional degrees of freedom. Because of this one chooses for each rectangle ~ = (Xli, X2i) X (YH, Y2i) two further basis functions

which vanish at the vertices of ~, so that continuity is ensured there if bP) and b~2) are extended by zero outside ~. The first problem which arises for non-conformal elements is the definition of the stiffness matrix L. A definition in terms of Lij = a(bj , bi ), as in (1.8a), is not possible because bj, bi ¢ VI Therefore, in addition to discretising V to Vh, one must replace a(·,·) by a bilinear form ah (-, .): Vh x Vh - IR. For a partition of il into rectangles 14 one defines ah(',') by ah(u,v) :=

I: 1 I: i

aaf3(x)(Dau) (Df3u ) dx,

Ro lal.If3I:5m

if a(·,·) is given by (7.2.6). Notice that a(u,v) = ah(u,v) for u,v E V. Conformal finite-element methods always give consistent discretisations. For nonconformal methods one does not always have consistency. For this problem, in particular for the so-called "patch test" see Strang-Fix [1, p. 174], Ciarlet [1, §4.2], Stummel [2], Gladwell-Wait [1, p. 83-92], and Thomasset [I]. That an ansatz which seems completely reasonable can possibly yield completely false discretisations is shown by Exercise 8.7.1. Let". be an admissible triangulation. Assume that Vh := {u constant on each n E ".}. Show that (a) Vh rt. V := Hl(il). (b) inf{lu - vlo: v E Vh} ~ Chluh for all u E V = Hl(n). (c) Let a(u, v) := fn(''lu, Vv) dx and ah(u, v) := L,T.ET fT. (Vu, Vv) dx. What is the result for Lij = ah(bj , bi )?

Non-conformal elements are, in particular, of importance for equations of higher order, since they are less complicated than conformal elements. 8.7.2 The Trefftz Method

In Theorem 6.5.12 it was shown that for symmetric and V-elliptic bilinear forms the weak formulation (1.1) of the boundary value problem is equivalent to the variational problem Find u E V with J(u)

= min{J(v):v E V}

(8.7.1)

(cf. also Theorem 7.2.9, Exercise 7.3.8). Another variational problem Find W E W with K(w)

= max{K(w):w E W}

(8.7.2)

is called the dual or complementary variational problem to (1), if both have the same solutions u = w E V n W.

8.7 Additional Remarks

201

For example, the Poisson problem, -Llu = f E L2(D) in D, u = 0 on r, leads to J(v) = fa IVvl2dx for v E V := HJ(il). A special dual variational problem is due to Trefftz (1926):

K(w) := -folVwl2 dx for w E W := {w E Hl(D): -Llw = f} (cl. Velte [1, p. 91]). Notice that v E V satisfies the boundary condition v = 0 on r, while for w E W no boundary condition is required though it must satisfy the differential equation Lw = f. 8.7.3 Finite-Element Methods for Singular Solutions The error estimates provided so far, such as for instance (4.7), depend on the assumption u E H2(D) n V, which will be discussed more carefully in Section 9.1. That this assumption does not always hold is shown by Example 8.7.2. (a) The Laplace equation Llu = 0 in the L-shaped region of Example 2.1.4 has the solution u = r2/3 sin«2 0, (4), m

= 1, and (8).

(8.8.11)

The ideas involved in the proof can, in principle, be carried over to the case 2m > 2, i.e., to boundary value problems of higher order. The inverse estimate becomes (8.8.12)

In order that inequality (4) hold one must be careful in defining the norm 1I·lIh. If all the components Ui of u are nodal values (Pu)(xi ) (as for instance is the case for the spline ansatz for Vh C H2(n)), then 1I·lIh can be defined as in (2b). However as soon as the components Ui of u involve derivatives (Dapu)(xi ), the u1 in (2b) must be replaced by (hlalui)2. For example, when the Hermite functions of (5.5) are used then u has the components Uti and U2i (cf. Exercise 8.5.3), where Uti = (Pu)(xi ) and U2i = .t~ (Pu)(xi ). The appropriate definition of 1I·lIh is lIull~ := hn L:i(U~i + h2u2i) with n = 1, since (0,1) C ]Ri. Exercise 8.8.9. Let V cUe V' be a Gelfand triple and assume Vh C V. Show that the inverse estimate (8.8.13a)

implies

9 Regularity

9.1 Solutions of the Boundary Value Problem in H8(lJ), 8> m 9.1.1 The Regularity Problem The weak formulation of a boundary value problem

Lu = 9 in

n,

Bu =

s+n/2 ~ n/2. Then the weak solution 1£ of (7) belongs to C·(Rn ). Hence for ~ ~ 2m, the weak solution is also a classical solution.

PROOF. The statement results from Sobolev's lemma (Theorem 6.2.30). •

9.1 Solutions of the Boundary Value Problem in HB(O), s

>m

213

Corollary 9.1.7. Let a(.,.) be Hm(ffin)-coercive. Let the conditions (6) and IE H1c-m(llln) be satisfied lor all k E IN. Then the weak solution of problem (7) belongs to coo(ffin). The conditions are satisfied in particular if I belongs to C~(ffin) and the coefficients aa~ are constant. The generalisation of u E H m+1c(ffin) with k E lN to u E Hm+s(ffin) with real s > 0 reads as follows:

Theorem 9.1.8. Let a(.,.) in (5) be Hm(ffin)-coercive. Let s= k

+ e,

0 < e < {) < 1,

k E lN u {O},

For the coefficients (n

= ffin)

aa~ E ct+I~I-m(n)

for k

t:= k

+ {).

let

+ 1,81 ~ m,

aa~ E LOO(n) otherwise. (9.1.12)

Then each weak solution of (7) with IE H-m+s(ffin ) belongs to Hm+s(ffin ), and satisfies the estimate

(9.1.13)

PROOF. The beginning of the induction is given by k = 0, i.e., 0 < s < t < 1. Replace the difference quotient 8 = 8h,j by an approximation to its power 8s :

Ru(x) := Rh,jU(X) := h- s

f:

,,=0

e-"h( -I)" (S)u(X + JLhej) , I-'

where ej is the jth unit vector. Here (~) = 1, and

G) (-1)" = -8(1 - 8)(2 - 8) ...

(I-' -1 - 8)/ JL!

is the binomial coefficient.

Exercise 9.1.9. Let 0
O,s~

H,J, .. ·,m-!}. The norm 1·ls of H8 (JR+.)

is equivalent to

111'1.1.1118 ;= (1'1.1.15 +

L

IDaul~_m)1/2.

(9.1.15)

lal=m

PROOF. The relatively elementary case s

~

m is left to the reader. For

o < s < m too, the proof would be considerably simpler if in (15) one were to

replace the dual norm 1·ls-m of (Ho-S(R+'))' by that of (Hm-s(JRi.))'. Step 1. First we prove the statement for {l = JR" instead of {l = JR+.. According to Theorem 6.2.25a, for {l = JR" the norm III· Ills is equivalent to IlIulll~

;=

(1'1.1.1& +

L

(IDaul~_m)2)1/2.

lal=m

Since (llIulll~)2 =

I lR" [1 + CLlal=m lea I2 )(1 + leI 2 y- mllll(e)1 2 de and

o < Go(l + lel 2 )S ::; 1 + ( L lea I2)(1 + leI 2 )s-m ::; G1(1 + lel 2 y, lal=m

III . 1I1~ and I 'I~ are also equivalent. Step 2. For the transition to (l = JR~ the following extension HS(JR") must be investigated, where x = (x', x,,) E JR";

~; HI! (JR~) -+

216

9 Regularity

(¢U)(x) := u(x)

x E R+,

for L

(4Ju)(x',x n ) :=

L av[u(x', -lIXn) + u(x', -Xn/lI)]

for Xn

< O.

1'=1

We list the properties of 4J as

Exercise 9.1.13. Let the coefficients av of 4J be selected as the solution of the system of equations E~=l av ( lIA: + II-A:) = (_I)A: (0 :::; k :::; L - 1). Show that (a) u E CL-1(R+) yields 4Ju E c L- 1(Rn). (b) 4J E L(HA:(R+), HA:(Rn» for k = 0, 1, ... ,L. (c) The operator adjoint to 4J reads L

(4J*u)(x', xn)

= u(x', xn)+ ~ av [~u(x"

-xn/lI)+lIu(x', -lIXn)]

for Xn

> O.

(d) (8/8xn )A:(4J*u)(x', 0) = 0 for k = 0, 1, ... , L - 2 and u E cA:(Rn). (e) 4J* E L(HA:(Rn), H3(R+» for k = 0,1, ... , L - 1. Hint: cf. Corollary 6.2.43. (f) 4J E L(H8(R+), H8(Rn» for 8 = 1 - L, 2 - L, . .. ,0,1, ... ,L. Step 3. 111·111" :::; OI·IB results from Da E L(H8(R+), H,,-m(R+» (provable via continuation arguments from Remark 6.3.14b) so that 1·1. :::; 11I·llIs remains to be shown. Step 4. luis:::; 14Ju1" :::; 1l1¢U1l1. is true according to Step 1 of the proof. The inequality 1l1¢U11l. :::; Cillulill., which would finish the proof, reduces to

IDa¢UI,,_m :::; OIDaul s _ m for Let

lad = m,

'1.£

E

(9.1.16)

H 8 (lll+).

lal = m. For

(4Jau )(x', xn) := u{x', xn) { E~=l av [( _lI)a nu (x', -lIXn ) + (_ttn u(x', -xn/lI)]

for Xn > 0, otherwise,

one verifies Da4J = 4JaDa. As in Exercise 13f one shows that 4Ja E L(H8(lll+), HB(llln» for 8 = 1 + m - L, 2 + m - L, ... , L - m. This result can be carried 8 E IN (Le., over to real 8 E [1 + m - L, L - m] except for the cases 8 = -1/2, -3/2, ... ) (cf. Lions-Magenes [1, p. 54 ff]). Let L ~ 2m + 1 and v E Hm-"(llln). Since 4J~v E HO"-S(lll+), one infers from (D a4Ju, V)L2(lRn) = (Dau, 4J~V)L2(IR~P the estimate

!-

I{Da¢U, v)ol

= IDaul,,-mll4J~IIH;'-'(R~jJ+-Hm-'(lRn)lvlm-8'

for all v E Hm-8(Rn ), and therefore (16) with C := 114J~IIH;'-'(IR+)+-Hm-'(lRn) = l14>aIlH·-m(lRn)+-H·-m(IR+)'

9.1 Solutions of the Boundary Value Problem in HS(G),

8

>m

217

• PROOF. (of Theorem 11) (a) First let k = 8 = 1. The proof of Theorem 3 can be repeated for the differences ah,jU (j = 1,···, n - 1) and implies the existence of the derivatives au/aXj E Hon(IR+'}, j f n. Thus one has Dau E Hi(IR+.) for all lal = m except for a = (0, ... ,0, m). (b) We set a:= (O, ... ,O,m) E 7l.n, w:= a&aDau,

Fa(v):=

r w(x)Dav(x)dx for lal = m,

1JR'-+-

where a&a is the coefficient from the bilinear form (5). The remainder of the proof runs as follows. In part (c) we will show that (9.1.17)

= (w,Dav)o = (-I)m(Da w,v)o, (17) means that Daw E Hi-m(IR+.) and IDaw\t-m ~ C for lal = m. According to Lemma 12 it follows that w E Hl(IR+.). The coercivity of a(·,·) implies uniform ellipticity of L = Elal=I.BI=m(-I)mD.Baa.BDa, i.e., Eaa.B~a+.B ~ €1~12m (cf. Theorem 7.2.13). For ~ = (0,··· ,0, 1) one obtains a&&(x) ~ €. Hence it follows from wE Hl(IR+.} and D'Yaaa E LOO(nt+'), hi = 1, that D&u E Hl(nt+'). According to part (a) all other derivatives Dau (Ial ~ m, a f a) belong to Hl(nt+.) anyway so that u E H m - i (nt+') has been proved. (c) Proof of (17). For each a f a there exists a "Y with bl = 1, "Yn = 0, ~ "Y ~ a (component-wise inequalities). Integration by parts yields Since Fa(v)

°

for v E CO' (IR+'); and thus

IFa(v)l

~

Calvlm-l with Ca := C[\ulm + ID'Yulml for all v E CO'(IR+').

Here we used the fact that D'Yu E Hm(IR+') according to part (a). Since CO'(IR+') is dense in no-1(IR+'), (17) follows for a f a. There remains to investigate a =

Fa(v)

= a(u, v) -

a. We write

a(u, v) with a(u, v)

=~'

r aa.B (DCtu) (D.Bv ) dx,

1JRn+

where E' represents the summation over all pairs (a, P) (a, P) f (a, a) there exists a "Y with

hl=l,

O~"Y~P,

f

(a, a). For each

a+"Yf(O, ... ,O,m+l).

As above, one integrates each summand with

IPI = m by parts:

218

9 Regularity

for v E CQ'(IR+'), and estimates using C\V\m-l with C = C(u). Altogether one obtains \a(u,v)\ ~ C\V\m-l. Together with \a(u,v)\ = \(f,v)o\ ~ \f\-m+l\v\m-l, (17) also follows for a = a. (d) By induction for (k = 2, ... ) one proves in the same way \Fa(v)\ ~ Clvlm-k, and from this u E Hk+m(IR+'). For real s > 0, s -I- 1/2,···, m-1/2, one proves correspondingly \Fa(v) \ ~ Clvl m- B and hence u E HB+m(IR+'). • The generalisation of Theorem 11 to inhomogeneous boundary values reads as follows.

Theorem 9.1.14. Let the bilinear fonn a(·,·) from (5) be Hir(IR+.)-coercive. For an s > 0, s ¢ {1/2, ... , m - 1/2} either let (6) hold if s = k E 1N, or (12), if s ¢ 1N. Let u E Hm(IR+') be the weak solution of the inhomogeneous Dirichlet problem

= (f,v)o for all v E non(IR+'), alu/ an' = CPI on r = aIR+ for I = 0, 1, ... , m - 1, a(u,v)

(9.l.18a) (9.l.18b)

where f E H-m+B(IR+'),

CPI E Hm+B-I-l/2(r)

(0 ~ I ~ m - 1). (9.1.19)

Then u belongs to Hm+B(IR+') and satisfies the inequality (9.1.20)

PROOF. For m = 1 Theorem 6.2.32 guarantees the existence of Uo E Hm+B(IR+') which satisfies the boundary conditions (18b) (for m > 1 cf. Wloka [1, Theorem 8.8]). w := u - ua is the solution of the homogeneous problem a(w,v) = F(v) := (f,v)o - a(ua,v) (cf. Remark 7.3.2). Theorem 11 may also be carried over to the right-hand side F( v) under discussion here (instead of (f, v)o) and yields w E Hm+B(IR+'). • By similar means one proves

Theorem 9.1.15. (Natural boundary conditions) Let the bilinear form a(·,·) from (5) be Hl(IR+.)-coercive. For s > 0 either let (6) hold if s = k E 1N, or (12) if s ¢ 1N. Let u E Hl(IR+.) be the weak solution of the problem a(u, v)

= f(v):= f g(x)v(x) dx + f cp(x)v(x) dr for all v E Hl(IR+.),

lR+

lr

9.1 Solutions of the Boundary Value Problem in H"(a),

8

>m

219

where gEH

s-1

:=

{HS-l(IR~)

(H1-s(IR+.»)'

for s ~ I} E Hs-1/2(r). for s < 1 • m - 1. aa{J E LOO(Jl) otherwise. if t ¢ IN.

Then each weak solution u a(u, v)

=

e Hrf(Jl) of the problem

In

f(x)v(x) dx

for all v E H[)({l)

with f E H-m+s (Jl) belongs to Hm+s (Jl) n Hrf (Jl) and satisfies the estimate (9.1.23)

For inhomogeneous boundary conditions 8'u/8nl

= HA:+~+n/2(fl) for u to be a classical solution from ck+~(n). The minimal conditions for u E CA:+~{n) result from a different theoretical approach, which essentially goes back to Schauder. The following theorem, for example, can be found in Miranda [1, §V]. It shows the Ck+~_ regularity of the operator Lin (5.1.1).

Let k ~ 2, 0 < A < 1. Let fl E Ck+~ be a bounded domain. Let the differential operntor L = E Q.jj{)2/{)x.{)Xj + E Q.j{)/{)Xi +a be uniformly elliptic in n (i.e., (5.1.3a) holds). Let Q.jj, as, a E Ck-2+~(fl) and f E Ck-2+~(n),

m

223

The proof of Theorem 21 uses an isomorphism R = R* related to Rh,j (cf. proof of Theorem 8) between HO+8(il) and Hlf(il) and also between Hlf(il) and Ho - 8(n) such that the form b(u,v) := a(Ru,Rv) is Ho+S(il)-coercive. It is necessary to prove that btu, v). := a(u, R 2v) is also Ho+S(il)-coercive. We know f e H-m+s(il) implies f := R2f e H-m-s(il). Each solution of a(u, v) = (f, v)o is also a solution of a(u, R 2v) = btu, v) = (i, v)o = (f, R 2v)o so that u e HO+8(il) follows.

9.1.5 Regularity for Convex Domains and Domains with Corners

A domain n is convex if with x',x" e il, x' +t(x" -x') belongs to il for all

o :5 t :5 1 . Convex domains in particular belong to CO,l, but permit stronger regularity statements than Theorem 21.

Theorem 9.1.22. (Kadlec [1]) Let il be bounded and convex. Let the bilinear form (5) be Hlf(n)-coercive. Let the coefficients of the principal part be Lipschitz-continuous: aafj

e CO,l(G)

for allial

= 1.81 = 1;

for the remaining ones let the following hold: D"Yaa{3eLOO(il)

foralla,.8,"'(

with"'(:5I.B1, lal+I.8I 8. Let u eVe Hm(il) be a weak solution of the problem a(u,v) = fa1vdx for all v e V, where the restriction fla1 belongs to Hm+8(ilo) belongs to H-m+8(ild. Then the restriction ofu to and satisfies

no

PROOF. A special covering of il is given by UO = ill. U l = il\ilo. Thus we obtain the assertion from Part b of the proof of Theorem 16. •

9.2 Regularity Properties of Difference Equations The convergence estimates for difference equations in Section 4.4 read, for example, lIu - uhlloo ~ Ch2I1ullc4(o) under the condition that u e C4(D) or u e ca·l(.'O). This regularity assumption is frequently not satisfied (cf. Examples 2.1.3-4). A comparison with the error estimate lu - uhlo ~ Ch21ul2 for the finite-element method suggests that similar estimates also exist for difference solutions. To obtain the latter, one needs to replace the stability estimate IILh"11l2 ~ C (or IILh"llloo ~ C), which corresponds to L-l e L(L2(il), L2(il», by stronger estimates which correspond to L -1 e L(H-l(il), HJ(il» or L-l e L(L2(il), H2(il».

9.2.1 Discrete Hl-Regularity For il C IRn the following grids are defined:

Qh:= {x e IRn:x,

= lIih,

IIi

e Z},

ilh:= ilnQh.

A grid function Vh defined on ilh is extended to Qh by Vh

Vh(X)

=0

for x

e Qh \ilh.

(9.2.1)

= 0: (9.2.2)

The Euclidean norm is now called the L~-norm: (9.2.3a)

9.2 Regularity Properties of Difference Equations

227

It comes from the scalar product

L: Vh(X)Wh(X).

(Vh,Wh)O:= (Vh,Wh)L~ := hn

(9.2.3b)

XEQh

The discrete analogue of HJ(n) is Hl with the norm IVh!t :=

IIVhIIH~ := [IVhl~ + ~ latvhl~] 1/2

where at is the forward difference in the

Xi

t

= 0 on Qh \nh)

(9.2.3c)

direction. The dual norm reads

The associated matrix norms IILhIlHl+-H-1 h h ILhit+-o, etc., are defined by ILhl i +-j := SUP{ILhVhldlvhlj: 0

(Vh

=

ILh!t+--b IILhIlHl+-L2 = h h

Vh satisfies (2)}

for i,j E {-I, 0, I}. (9.2.3d')

Exercise 9.2.1. Show that (a) ILhlo+-o is the spectral norm of Lh (cf. §4.3). (b) The following inverse estimates hold:

(9.2.3e) The matrix Lh yields the bilinear form ah(uh, Vh) := (LhUh, Vh)L2. h

(9.2.4a)

ah(-, .) is said to be Hl-elliptic if aCE> 0 exists such that ah(uh, Uh)

2 CEluhl~ for all Uh and all h > O.

(9.2.4b)

Correspondingly, ah(-,') is said to be Hl-coercive if there exist CE > 0 and C K E lR with ah(Uh, Uh)

2 CEluhl~ - CKluhl~ for all Uh and all h > O.

(9.2.4c)

As defined in Sect. 4.5, Lh (resp. ah(-,'» is said to be L~-stable if

IL;llo+-o ::; Co

for all h > O.

(9.2.4d)

We call Lh Hl-regular if IL;111+-_1 ::; C 1 for all h

> O.

Exercise 9.2.2. (a) Hl-regularity implies L~-stability. (b) IAhl i +-j = IAII-j+--i for i,j E {-I,O, I}. (c) If Lh is Hl-regular, then so is LI.

(9:2.4e)

228

9 Regularity

(d) If Lh and LI are stable with respect to 1·100, Le., IILhlloo :::; Goo, IILIlloo :::; Gl, then L~-stability (4d) follows with Go := (G1Goo )1/2. (e) Hl-ellipticity implies Hl-regularity. The following statement resembles the alternative in Theorem 6.5.15. Theorem 9.2.3. II ah(-,·) is Hl-coercive and il Lh. is L~ -stable, then Lh. is also Hl-regular. PROOF. (a) Let LIuh Coercivity provides

= III. such that ah(Uh.,uh) = (uh,LIuh)O = (uh,lh)O.

IUh.l~ :::; [all. (Uh' Uh) + Gluh.l~lIGE

= [(Uh, A)o + Gluhl~lIGE

:::; G'lIlh.lo + Gluh.lollUh.lo. On the basis of the stability estimate IUh.lo :::; GolAlo we obtain IUhl~ :::; GffiAla. From this one infers ILr- 1h+-o :::; ,fC" := G* and hence IL;;-110+-_1 :::; C* (cf. Exercise 2b). (b) Now let LhUh = III.. According to Part a, one has IUhlo :::; G*llhl-1. By estimating lah(uh,uh)1 = l(fh,Uh)ol :::; IAI-lluhl1 through !GEluhl~ + !GElllhl~lt one obtains the coercivity

•

'Instead of the L~-stability in Theorem 3 one can also assume the solvability of the continuous problem and a consistency condition (cf. Corollary 11.3.5).

By analogy with Lemma 7.2.12 the following may be proved. Exercise 9.2.4. If Lh is Hl-coercive, and if 6Lh := ao + Li bia; + Li a; O. Further, let aij E CO(n) hold. Show that for sufficiently small h the associated matrix Lh is H~-regular; for all h > 0, Lh is H~-coercive. Hint: For sufficiently small h the following holds: -au (x

+~, y )d~ - a22 ( x,y + ~)d~ -

[a12(x+~,Y+h) +a12(x+~,Y)]dld2

~ ~(4 + 4), for all dt, d2 E JR. The H~-coercive difference methods constructed so far remain H~-coercive if differences of lower order are added (cf. Exercise 4) or if the principal term (auuz)z + '" is replaced by allu zz + .,. with all E Cl(n). The above difference methods are described by the same difference operator regardless whether the grid points are close to or far from the boundary. The homogeneous Dirichlet boundary condition is discretised by (2): "Uh = 0 on Qh \ fih " . If one wants to approximate the boundary condition more accurately, one needs to select special discretisations in the points near the boundary of fih (cf. Section 4.8.1,4.8.2). One thus obtains an irregularity which makes a proof of H~-regularity, as in Example 5, difficult. We begin with the one-dimensional case. Lemma 9.2.7. Let Lh be the matrix of the one-dimensional Shortley- Weller discretisations of -u" = f on fi, U = 0 on 8fi:

230

9 Regularity

PROOF. (a) First, let n be assumed to be connected. Let the grid points of nh be x, := Xo + lh E n for l = 0, ... ,k > o. Let the boundary points of n be

Xo - s',oh and Xk + Sr,kh with s',o, Sr,k E (0,1). The other factors of Equation (7) in x = Xi are S',i = sr,; = 1. If one takes into account Vh = 0 on ffi\n one obtains the following identity: k

(Vh, LhVh)O := h

L vh(x,)(LhVh)(X,) '=0

= 18+Vhl~ +

+ Vh(Xk)

h-

1{

Vh(XO) [ (~ -

[(~ -

2) Vh(Xk)

+

2) Vh(XO) +

(1 -1~ S,) Vh(Xl)]

(1 -1~ sr) Vh(Xk-l)]} ,

(9.2.8a)

where s, := s',o and Sr := Sr,k. Since Vh(XO) = M+Vh(X-l) and Vh(Xl) = h[8+Vh(X-l) + 8+Vh(XO)) because of Vh(X-t} = 0, the first summand inside the braces can be written as

(9.2.8b)

For a := 8+Vh(X-tl, f3 := 8+Vh(XO) use the inequality -af3 ~ _ja 2 - Af32 with j = (s, + 2)/ s, and note that the function s(1 - S)/[4(1 + S)(2 + s)] is bounded by 0.018 (maximum at S = (y'3 -1)/2). The second summand inside the braces is treated analogously and yields

(Vh' LhVh)O ~ 18+Vhl~

-

0.018h[8+vh(xo)2 + a+Vh(Xk_d 2).

(b) In Part (a) it was assumed that k

(Vh' LhVh)O =

(9.2.8c)

> o. For k = 0 one obtains

h-l~Vh(XO)2 = ~[8+Vh(xo)2 + a+Vh(X_t}2 ~ la+Vhl~, S'Sr

S'Sr

so that (8c) also holds. (c) Let n be arbitrary. Let the components of connection be Ii = (ai, b;) (i E 7l) with bi ::; a;+1. Let Lr) be the discretisation matrix associated with h Each Vh with support in nh = Qh n n can be written as Li vii) where

9.2 Regularity Properties of Difference Equations

Vr) nk

= 0 outside Ii. Let x&i) and

i ) :=

231

X~(i) be the first and last grid points of

nil. n Ii. The statement follows from 18+Vhl~ ~ L{\8+v~)I~ + 1h8+vii)(x~(i)]2} = 2L 18+v~)I~ i

i

• n

c m.2 be bounded. Let Lh be the matrix associated with the Shortley- Weller discretisation of the Poisson equation (cf Section 4.8.1). Then Lh is H~-regular. Theorem 9.2.8. Let

PROOF. Let L~ ILK] be the portion of the x differences [y differences] such that Lh = L~ + LK. The restriction of L~ to a "grid row" {(I/h, y) E n: 1/ E Z} (y is fixed) corresponds to the matrix Lh in Lemma 7. Thus one obtains (VII., L~Vh)O ~ il8tvhl5. With the analogous inequality (VII., LKvh)o ~ !1 8t v h15 one obtains 2(Vh' LhVh)O ~ 18tvhl5 + 18tvhl5 = IVhl~ - IVhl5 ~ Enlvhl~, thus IIL;;-1Ih O. One obtains an analogous estimate for the difference method in Section 4.8.2. • 9.2.2 Consistency

In the following we carry over the estimate lu h - 1.£11 S Chlu\2 S C'hill o, which holds for finite-element solutions, to difference methods. To this end one needs to prove the consistency condition

ILhRh - Rh L I-l+-2 := IILhRh - RhLIIHh"1+-H2(0) S CKh

(9.2.9)

for suitable restrictions

Rh: H2(il)

n HJ(il) --+ H~, R h: L2(il) --+ L~.

To construct the restrictions we first continue 1.£ E H2(il) in ii := EJu E H2(IR?). According to Theorem 6.2.40c one assumes (9.2.lOa) for all 1.£ E with

H2(il) n HJ(il). An analogous continuation Eo: L2(il) --+ L2(IR?)

7 := Eol =

I

on il,

IIEoIIIL2(JR2) S CIIIII£2(I1)

is given, for example, bl, 7 = 0 on IR?\il. Uh' uK: Ctf(JR.2 ) --+ Co(JR. ) be defined by

(UhU)(X, y) = h- 1 f

h/ 2

u(x+€, y) de,

for all IE L2(il) (9.2.lOb) Let the averaging operators

(uKu)(x, y) = h- 1

-~

fh/2 -~

u(x, Y+17) d17· (9.2.11)

The restrictions Rh, Rh are chosen as follows:

Rh := uhu~E2'

Le., (RhU)(X, y)

= h- 2 f

with u = EJu for (x, y) E ilh, Rh := (UhU~)2Eo, Le., (Rhf)(x,y)

h/2 fh/2 fh/2 fh/2

h/2 fh/2 -h/2 -h/2

u(x +~, y + 17) de d17 (9.2.12a)

7(x + €+ ~', y + 17 + 17') de de' d17 d17' -h/2 -h/2 -h/2 -h/2 with I = Eo! for (x, y) E ilh . (9.2.12b)

= h- 4 f

9.2 Regularity Properties of Difference Equations

233

The characteristic properties of the convolutions u~,u~ are the subject of Exercise 9.2.12. Let 8:IJ be the symmetric difference operator (8:IJu) (x, y) := [u(x + hj2, y) - u(x - hj2, y)]jhj 811 is defined analogously. Show that fl:1I_1I:IJ 8_8:IJ_:lJ8 8_811_118 () a UhUh - UhUh' U:IJ - 7fi Uh - Uh7fi' u1I - 8ijUh - Uh8ij' (b)lIu~IIH"(R2)""'H"(1R2) ~ C, lIu~IIH"(R2)+-H"(1R2) ~ C (in particular for k = O,±1,2). (c) lIu~u~vIlL~ := [EXEOh l(u~u~v)(x)12P/2 ~ IIvll£2(R2) for all v E L2(:m?). (d) For a E ao,l(:m?) holds lIauhu~ - u~uKaIlL~+-L2(R2) ~ Chllallco.l(1R2). Here (au~u)(x) = (a(x)uhu)(x) and (uhau)(x) = (uh(au»(x). (e) lIa(u~)V(Uh)1' - (Uh)V(uh)l'aIlL~+-£2(1R2) ~ CVl'hllallco.l(1R2) for v, JL E IN. (f) 111.1. - (uht(Uh)l'uIlH"(R2) ~ CVl'hliuIlH"+1(1R2) for 1.1. E Hk+l(:m?). Consider the differential operator n

L

=

n

L a.j(x){PjaxiaXj + La.(X)ajaxi + a(x). i,j=l i=l

(9.2.13a)

First we assume that n = :m.n and discretise L with the regular difference operator (9.2.13b) Lh = La.j(x)a!a;' + a. (x)a! + a(x) with an arbitrary combination of the ±-signs. Lemma 9.2.13. Let n = :m.n . Let a.j,ai,a E ao· 1(:m.n ). Let Land Lh be given by (13a,b). Then the consistency estimate (9) holds. PROOF. To simplify the notation let us assume that n Rhauu:IJ:IJ

= 2. Let

= (uhu~)2allu:IJ:IJ = all(x)(uhU~)2u:IJ:IJ - 01

with IlolllL~ ~ Chllal1l1co.l(1R2)luI2 (cf. Exercise 12e,b). Let the term corresponding to allUz:IJ in (13b) be, for example, all(Xj h)a,tat. Exercise 12a shows that

with 02(X, y) := -a,tu~uX[u(x - h, y) - u(x, y)] = -UhU~U~[U:IJ(x - h, y) U:IJ(x, y)] and 1102112 ~ lIuK[U:IJ(X - h, y) -U:IJ(X, y)]lIL2(1R2) ~ hluI2 (cf. Exercise 12c,b). Since 8:IJ8:IJ = a:a;;, the error term 02 does not appear if one also approximates a11U:lJfI: by a11(X)ata;;. Finally one obtains au a; a; u~ u~ 1.1. = all at a; Rh 1.1. + all a; a; [U~ u~ - u~ uK)u = a11 a;a; Rhu + alla;UhU~[uK - uh]u:IJ = alla;a; Rhu - all a;03

234

9 Regularity

with 115s1lL~ = II(}"~(}"~[(}"~ - (}"~]U:eIlL~ ~ II[(}"~ - (}"~]uxlIL2(R2) ~ Chlul2 (cf. Exercise 12c,f). Putting this altogether one obtains

82 ] _ [an 8x± 8x± Rh - Rhau 8x2 u

= 51 + au8x± (52 + 5s).

For the first error term the following holds: (9.2.14a) For arbitrary Vh E H~ we have (Vh, a118;(~ + 5s»L2h

= -(8; [auvh], ~ + 5sh2. h

co· 1 (m?)

(9.2.14b)

it follows that 118; [avhlllL~ ~ lIallco'1(R2)lIvhIlL~ lIallco(R2)lIvhIlH~ ~ CllvhIlH~' so that

From a E

lIa118;(~

+ 5s)II H - = 1

h

+

sup I(Vh, au8;(52 + 6s»L21

tlhEH~

h

(9.2.14c)

~ CII 62 + 5s1lL~ ~ C'hluI2'

From (14a,c) we have

± ±

lIa ll8x 8z Rh -

_ 82 Rhall8x2I1Hh"l+-H2(R2) ~ Ch.

(9.2.14d)

Analogously, one shows

± ± lIa;j8x,8xjRh - Rha;j 8 2 /8xi 8xjIlHh"l+-H2(JR2) ~ Ch.

(9.2.14e)

For a;8;'Rh -Rha;8/8xi similar reasoning results in an O(h)-estimate for the norms II . IIHh"l+-Hl (R2) and II ·IIL~+-H2(R2). Both are upper bounds for the larger norm II . IIHh"l +-H2(m,2) such that (9.2.14f) Likewise lI aRh -

RhaII Hh"1+-H2(R2) ~ Ch.

Statement (9) follows from (14e,f,g).

(9.2.14g)

•

When generalising the consistency estimate to more general domains n c m?, the following difficulty arises. The entries of the matrix Lh according to (4.8.7) or (14.8.16) are not bounded by Ch- 2 • Rather, at points near the boundary the inverse of the distance to the boundary point enters, and this distance may be arbitrarily small. One way around this, would be to formulate the discretisation so that the distances between boundary points and points near near the boundary remain, for example, ~ h/2. A second possibility would be a suitable definition of Rh so that the product LhRh appearing in

9.2 Regularity Properties of Difference Equations

235

(9) can be estimated (cf. Hackbusch [2]). Here we choose a third option: Lh is replaced by the re-scaled matrix Lh = DhLh from Corollary 11.

Theorem 9.2.14. Let n E C 2 (or convex) and bounded. For the discretisation of Lu = f for L = -.6 on il with u = 0 on r use the discretisation LhUh according to (4.8.7) or (4.8.16). Let Lh = DhLh be defined as in Corollary 11. Then the consistency estimate (9.2.15)

holds. Here, the matrix Lh from (4.8.7/16) is only taken as an example. The proof will show that the estimate (9), respectively (15), also holds for other Lh if (LhUh)(X), x near the boundary, represents a second difference. First, two lemmas are needed. Let 'Yh C nh be the set of points near the boundary. If Vh is a grid function defined on ilh then we denote by Vh h" the function

(vhh,,)(x) = Vh(X)

for x E 'Yh,

(vhh,,)(x) = 0 for x

E

ilh\'Yh'

Lemma 9.2.15. Let il E CO· l be bounded. Then there exists a C = C(n) independent of h, such that (9.2.16) PROOF. From il E CO· l follows: there exist numbers K E IN and ho > 0 such that for all x E 'Yh, with h ~ ho, not all grid points {x+(vh,JLh): -K ~ v, JL ~ K} lie in n. For each x E ilh select a pair (vo, Po) E 71. 2 with -K ~ Vo, JLo ~ K and x + (voh, JLoh) ¢ n. At this grid point x the functions w~'" (-K ~ v, JL ~ K) are defined by w~O"'O(x) := Vh(X), w~"'(x) := 0 for (v,JL) =1= (VO,JLo). Therefore E~"'=-K w~JA = vhh" is a decomposition with w~"'(x) = Vh(X) or w~JA(x) = o. Without loss of generality let us assume that v > 0 and JL > O. For x E nh we define the chain

xO = X, xl = X + (h, 0), ... ,

XV

=

X + (vh, 0),

x+ (vh,h) , '" , x +JA = x+ (vh,JLh). definition, either w~"'(x) = 0 holds or x v +'" ¢ n,

xv+! =

V

According to the = 0 (cf. (2». In both cases we have

w~"'(xv+"')

\wh"'(x)\ ~ \wh"'(Xo) - whJA(xv+",\

= h 1~[w~"'(xO) ~ h[\a;w~"'(xl)\

w~"'(Xl») + ~[w~"'(xl) - w~"'(x2») + ... 1 + \a';w~"'(x2)\ + ... + \a';Wh(XV )\

+ \a;wh"'(XV +1)\ + ... + \a;WhJJ(Xv+"')\J

i.e.,

236

9 Regularity

and thus Iw~l'lo ~ hUvl18tw~l'lo + IJLII~w~l'lo] ~ 2Khlw~l'h. Summation over v, JL yields estimate (16) for Vh l1'h = 2.., w~1' • • In most cases x E 'Yh already has a direct neighbour in JR2\.!2 of x E 'Yh such that (16) follows with C = ..;2. This holds in particular for convex domains. Lemma 9.2.16. Let Rh be defined by (12a). Let have I(Rh1£)(~,)1 ~ ChIlE21£IlH2(Kh/2(m

where K h / 2 (e):=

for all

~

satisfy (lOa). Then we

eE r,1£ E HJ(.!2) n H2(.!2), (9.2.17)

{e +x E JR2:x E (-~,~) x (-~, ~)}.

PROOF. (a) Let Q = (-!,!) x

Iv(O) -

(-!, i).

~ v(x)dxl ~ C

For v E Jl2(Q) one shows

k(v~:l) + 2v~1I + v~lI)dxdy,

since the leftrhand side vanishes for linear functions 1£(x, y) = a + fix + 'YY. The proof is similar to the one for (8.4.2). Transforming from Q to Qh := h 2h) x ( -2' h 2h) gIves . ( -2' (9.2.18) (b) Let eE

r. Statement (17) follows from (18) with v(x) := u(e + hx) =

(~1£)(e

+ hx),

since 1£(e) =

o.

•

PROOF. (of Theorem 14) (a) Inequality (15) is proved if

I(vh' [L~Rh - DhRhL]1£)ol ~ Ch for all Vh E H~ and 1£ E H2(ll) n HJ(ll) with is split into

IVhh = 11£12 =

(9.2.19a) 1. To this end Vh

In part (b) we show I(v~, [LhRh - DhRhL]1£)ol ~ C1h,

(9.2.19b)

The other steps of the proof, (c) and (d), yield

I(v", [LhRh - DhRhLJ1£)ol ~ C2 h ,

(9.2.19c)

such that (19a) with C = C 1 + C2 follows. (b) Lenuna 15 shows (9.2.19d)

9.2 Regularity Properties of Difference Equations

For v~ one obtains Iv~h ~ IVhh + Iv~h yields Iv~h ~ Ch-llv~lo ~ CCs, thus

237

= 1 + Iv~h. The inverse estimate (3e)

Iv~h ~ C4 •

(9.2.1ge)

Let Lh be the (regular) difference operator on the infinite grid

Qh = {(lIh, p,h): II, p, e 7l}. Since the support of v~ is ilh\'Yh, (V~,L~Wh)O = (v~, LhWh)O holds for all Wh· Furthermore, (v~, DhWh)O = (vK, Wh)O. This proves the first equality in

l(vK, [L~Rh - DhRhL]U)ol

l(vK, [LhRh -

RhL]U)ol ~ C5hlv~hlul2 ~ C5 hC4 C6 := C1h,

=

where u := E2U is the continuation of u to JR2. Further inequalities result from Lemma 13, (1ge), and (9.2.19f)

(cf. (lOa)). Theorem 6.2.40c guarantees the existence of an extension u with (19f) if n e C2. Another sufficient condition for (19f) is the convexity of il. (c) The left side of (19c) splits into (v~, L~RhU)O and (v~, DhRhLu)O. The first part is estimated in part (d). Exercise 12c and (19d) yield the second term I(v~, DhRhLu)ol ~ Iv~lolDhRhLulo ~ C3 hC'ILuio ~

C6hlui2 = C6 h.

(9.2.19g) (d) We set Wh := (L~RhU)hh. Since the support ofv~ is contained in 'Yh, we have (v~,L~RhU)O = (V~,Wh)O. L~ contains differences with respect to the x and y directions. Accordingly we write Wh = w h + w~. In the following we limit ourselves (i) to the Shortley-Weller discretisation, (ti) to the term wh and (ii) to the case that

xe'Yh,

xr=x+(8rh,O)enh (Le.,8r =1),

x':=x-(8,h,O)er.

The other cases should be treated analogously. We set

u := RhU = (7h(7~E2U e H2(JR2). The Shortley-Weller difference in the x direction reads

where dx is the diagonal element of Dh. Since in the equation L~ Uh = fh the variables Uh(e), E r (for example, = x'), have already been eliminated, w~ (x) has the form

e

e

:J)() = wh A:J)() 2dx A( ') X + h2 ( 8, 8, + 8 ) U x .

wh X

r

238

9 Regularity

The factor 2dx /[h 2s,(s, + sr)), due to the definition of D h , remains bounded by 4h- 2. From Lemma 16 and K h/ 2(X') C K 3h / 2(X) one infers IWk(X) - wh(x)1 ~ 4h- 2 Iu(x')I ~ Ch- 1 IluIlH2(K3h/2(X»'

The second divided difference Wh in x

wh(X,y)

= dx

(9.2.19h)

= (x, y) can be represented by

1:,.

g(t)UX 3:(x+t,y)dt

with

for 0 ~ t ~ h, 2(t - h)/(h2 + h 2s,), -2(s,h + t)/(h2s,(1 + s,)), for -s,h ~ t ~ 0, o otherwise. From this one infers g(t)

IWh(X, y)1

={

~ dx [l h

g2(t)

-hili

~ 2h- 1

[1: U~3:(X +

dt] 1/2 [l

h

u!:.,(x + t, y) dt] 1/2

-hili

t,y)

dt f/ ~ Ch- I1ii 2

1

(9.2.19i) IlH2(K,h/2(X»

(cf. (6.2.5a)). From (19h,i) and the corresponding estimate for w~ one obtains IWh(X)1 ~ C7h-1I1iiIlW(Kah/2(X»' such that

IWhl~ = h2

L

L

IWh(X)1 2 ~ C? lliill~2(K'h/2(X» ~ 9C?IIiiIl~2(R2) XE"Yh XE"Yh ~ (3C7 C6 )2 := Cl·

From this follows I(Vh,L~RhU)ol ~ l(vh,Wh)ol ~ IVhlolwhlo ~ CahCs:= Cgh.

(9.2.19j)

•

(19g) and (19j) yield the required inequality (19c).

Remark 9.2.17. The proof steps for Theorem 14 can be carried out in the same manner for more general difference equations (for example, with variable coefficients, as in Lemma 13).

9.2.3 Optimal Error Estimates In the following we compare the discrete solution Uh = LJ;1 fh with the restriction uh := RhU of the exact solution U = L- 1f. From the representation

Uh - Uh

= LJ;1fh - RhU = LJ; 1Uh - Rhf) + LJ; 1R hf = LJ; 1(fh - Rhf) - LJ; 1 (LhRh - RhL)u

one immediately obtains

- Rhu

(9.2.20)

9.2 Regularity Properties of Difference Equations

239

Theorem 9.2.18. Let u E H2(n) hold for the solution of Lu = f. Let the right-hand side fh of the discrete equation L",u", = f", be chosen so that (9.2.21) If, furthermore, L", is H~-regular, and if the consistency condition (9) holds, then u'" satisfies the error estimate

(9.2.22)

Corollary 9.2.19. (a) Inequality (21) is satisfied in particular if one chooses

in:= R",/. (b) The choice fh(x) := I(x) for x E il", (cf (4.2.6b» leads to (21) if I E CO,1(il) or IE H2(n). In these cases the following even holds: If", - R",/lo $; Cllf", - R"'flloo $; C'hllfll c o.l(ii»

resp.

Ifh -

-

Rh/lo $; Ch

2

1/12.

(9.2.23)

(c) In Theorem 18 one can replace the H~-regularity of Lh by that of Lh DhL", (cf Corollary 11), (9) by (15), and (21) by

=

(9.2.21') The proof of (23) is based on the inequality (18).

•

Error estimates of order O(h2) can be derived in the same way if one has consistency conditions of second order. These are, for example, (24a) or (24b): (9.2.24a) (9.2.24b)

Remark 9.2.20. If nil. = Qh (i.e., n = IR.2) or n = (x',x") X (y',y"), the inequalities (24a,b) can be shown in a way similar to Lemma 13. Example 9.2.21. The difference method in Example 4.5.8 shows quadratic convergence. This case can be analysed as follows. Using Remark 20 one shows (24a). In the following section we prove the H~-regularity IL;;112. E CU. By E(>') one denotes the subspace of all e E V which satisfy Equation (1)[ resp. (2a)]. E(>.) is called the eigenspace for >.. With E*(>') one denotes the corresponding eigenspace of Equation (2b). >. is called an eigenvalue if dimE(>') ~ 1. Theorems 6.5.15 and 7.2.14 already contain the following statements:

c L2(Q) be continuously, densely and compactly embedded (for example, let V = H{f'(D) with bounded D). Let a(-, .) be Vcoercive. Then the problems (2a,b) have countably many eigenvalues>. E CU which may only have an accumulation point at 00. For all >. E .)

= dimE*(>') < 00.

W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, DOI 10.1007/978-3-642-11490-8_11, © Springer-Verlag Berlin Heidelberg 2010

(11.1.3)

253

254

11 Eigenvalue Problems

Exercise 11.1.S. In addition to the assumptions of Theorem 2 let a(·,·) be symmetric. Show that all eigenvalues are real and problems (2a) and (2b) are identical so that E()") = E*()"). For ordinary differential equations of second order (Le., n c IR 1 , m = 1) it is known that all eigenvalues are simple: dim E()..) = 1. This statement is incorrect for partial differential equations as the following example shows.

Example 11.1.4. The Poisson equation -Lle = )..e in the rectangle (0, a) X (O,b), with Dirichlet boundary values e = 0 on r, has the eigenvalues ).. = (v7r/a)2 +(J.L7r /b)2 with V, JL E IN. The associated eigenfunction reads e(x, y) = eV,~(x, y) := sin(v7rx/a) sin(J.L7ry/b). In the case of the square a = b one obtains for 1/ =1= J.L eigenvalues ).. = )..v,I-' = )..1-',11, which have multiplicity at least 2 since eV'I-' and el-',v are linearly independent eigenfunctions for the same eigenvalue. A triple eigenvalue, for example, exists for a = b, ).. = 507r2 / a2 : E()") = span { e i ,7, e7 ,i, e5,5}.

The eigenfunctions e E E()"), by definition, belong to V. The regularity investigations of Section 9.1 immediately result in a stronger regularity.

Theorem 11.1.5. Let V = H{f(n) with m ;::: 1, or V = Hm(n) with m = 1. Under the assumptions oj Theorems 9.1.16 [resp. 9.1.17] we have E()")

c

Hm+lI(n).

PROOF. Along with a(·, .), a),(u, v) := a(u, v) - )..(u, v)o also satisfies the assumptions. Since a),(e, v) = 0 for e E E()..), v E V, the statement follows from Corollary 9.1.19. • Besides the standard form (2a) there are generalised eigenvalue problems. An example is the Steklov problem

-Lle = 0 in

n, Be/an =)..e

on

r,

whose variational formulation reads e E Hi(D), fo(Ve, Vv}dx = >-'frevdr (v E Hi(n». One can show that all eigenvalues are real and that the statements of Theorem 2 hold.

11.2 Finite Element Discretisation 11.2.1 Discretisation Let Vh C V be a (finite-element) subspace. The Ritz-Galerkin (resp. finiteelement) discretisations of the eigenvalue problems (1.2a,b) read

eh E Vh,

a(en,v) = )..h(en,v)o for all v E Vh,

(I1.2.1a)

11.2 Finite Element Discretisation

255

(11.2.1b) The discrete eigenspaces Eh(Ah), Ei.(Ah) are spanned by the solutions of the problems (la) [resp. (lb)]. As in Theorem 11.1.2, dimEh(Ah) = dimE;;(Ah) holds. If a(.,·) is symmetric, then Eh(Ah) = Ei.(Ah). As in Section 8.1, the formulation (la,b) can be transcribed into matrix notation. Remark 11.2.1. Let e and e* be the coefficient vectors for eh = Pe and e*h = Pe* (cf. (8.1.6)). The eigenvalue problems (la,b) are equivalent to (11.2.1') where the stiffness matrix L is defined as in Theorem 8.1.3 and M by (8.8.7). Since in general M #- I, (I') involves generalised eigenvalue problems. Exercise 11.2.2. Show that (a) M is positive definite and possesses a decomposition M = AT A (for example, A = Ml/2 or the Cholesky factor). (b) The first problem in (I') is equivalent to the ordinary eigenvalue problem La = Aha with L := (AT)-lLA- 1 , a = Ae. The second problem in (I') corresponds to LTa* = Aha* with a* = Ae*. When investigating convergence, one must watch out for the following difficulties: (i) A uniform approximation of all the eigenvalues and eigenfunctions by discrete eigenvalues and eigenfunctions is impossible since the infinitely many eigenvalues of (1.2a) are set against the only finitely many of (la). It is only possible to characterise a fixed eigenvalue A of (1.2a) as an accumulation point of discrete eigenvalues {Ah: h < O}, and to set up estimates for IA - Ahl. (ii) If A and Ah are the continuous, resp. discrete, eigenvalues then dimE(A) = dimEh(Ah) need not hold. It is preferable to limit oneself to the case of simple eigenvalues where dimE(A) = dim Eh(Ah) = 1. If dimE(A) = k > 1, it may well be that the multiple eigenvalue A is approximated by several discrete eigenvalues A~), i = 1,···, k: dimE(A) = E~=l dimEh(A~». The error estimates of IA - A~) 1 are then generally worse than for simple eigenvalues. Only for the mean value .x := E~=l A~) /k does one obtain the usual estimates (cf. Babuaka-Aziz [1, p. 338]). Exercise 11.2.S. Let the eigenvalue problem be as in Example 11.1.4 with a = b and A = 5071"2/ a2 • Let Vh consist of linear elements over a square grid triangulation. Show that to the given triple eigenvalue A corresponds a double eigenvalue A~l) and a single eigenvalue distinct from it, A~2), with limh......O A~) = A (i = 1,2). Hint: the nodal values of the discrete eigenfunctions agree with the continuous eigenfunctions e 1,7, e7,l, and e5 ,5.

256

11 Eigenvalue Problems

11.2.2 Qualitative Convergence Results This section concerns the question as to whether Ah -+ A and eh -+ e for h -+ O. The rate of convergence will be discussed in Section 11.2.3. The basic assumptions are the following: Let a(-.·): V x V

-+

lR be V-coercive,

(1l.2.2a)

c

L 2 (n) be continuously, densely, and compactly embedded, (1l.2.2b) so that the Riesz-Schauder theory is applicable (Theorem 11.1.2). Furthermore, let a sequence of subspaces Vh, (14 -+ 0) be given which increasingly approximate V (cf. (8.2.4a)): Let V

(l1.2.2c) We define a~(·,·): V x

V

-+

cr:,

W(A):=

a~(u, v) :=

a(u, v) - A(U, v)o,

(l1.2.3a)

la~(u, v)l,

(1l.2.3b)

inf

sup

uEV

tiE V

lIullv=l .I1t1l1v=l

Wh(A):=

inf

sup

UEVh

tlEVh

la~(u,v)l.

(l1.2.3c)

lIullv=l IItlllv=l

Exercise 11.2.4. Let L and Lh be the operators associated with a(·, .): V x V -+ lR and a(., .): Vh x Vh -+ lR (cf. (8.1.lOa)). Show that (i) If A is not an eigenvalue we have (11.2.4) (cf. Lemma 6.5.3 and Exercise 8.1.16). (ii) W(A) and Wh(A) are continuous in A E cr:. (iii) If in (3b,c) one replaces a~(u, v) by a~(v, 11.) one obtains the variables W* (A) and wh (A) which correspond to the adjoint problem. The following holds: W*(A) = W(A) and wh(A) = Wh(A). Hint: Use Lemma 6.5.17. With the aid of (4) and Theorem 6.5.15 one proves the following connection between W(A), Wh(A), and the eigenvalue problems.

Remark 11.2.5. Let (2a,b) hold. A is an eigenvalue of (1.2a) if and only if W(A) = 0, and Ah is eigenValue of (la) if and only if Wh(Ah) = o. Lemma 11.2.6. Let a(·,·) be V-coercive (cf. (2a)). Then there exists a J.L E lR such that aJ.'(-'·) is V-elliptic. In addition, then w(J.L) ~ l/CE and Wh(J.L) ~ l/CE hold with CE > 0 from Definition 6.5.13.

11.2 Finite Element Discretisation

257

Let A c CD be compact. Let (2a--c) hold. Then there exist numbers C > 0 and .,,(h) > 0, independent of A E A with limh-to .,,(h) = 0 such that

Lemma 11.2.7.

WheAl

~

weAl

CW(A) - .,,(h),

~

CWh(A) - .,,(h)

for all A E A. (11.2.5)

PROOF. Let the operators Z = ZeAl: V - V and Zh = Zh(A): V - Vh be defined as follows:

z := Z(A)U is the solution of al'(z, v) = (A - JL)(u, v)o for all v E V, (11.2.6a)

zh:= Zh(,x)U is the solution of al'(zh,v)

= (A -

JL)(u,v)o for all v E Vh , (1l.2.6b)

where JL is chosen according to Lemma 6. It shows

IIZlIv+-v' From a>.(u, v)

s Cz ,

IIZhllv+-vl S Cz

= al'(u - z, v)

w(A)lluliv

s

(11.2.6c)

and the definition of weAl one infers

la>.(u, v)1 =

sup

for all A E A.

sup

vEV

vEV

IIvllv=l

IIvllv=l

lal'(u - z, v)1

s Csllu - zllv

(11.2.6d) with Cs := IIL-JLlllvl+-v. For an arbitrary U E Vh one infers from a>.(u, v) = a,.(u - zh, v), Lemma 6, and (6d) that sup

la>.(u, v)1 =

vEV"

IIvllv=l

~

C;/lIlu -

sup

lal'(u - z\ v)1 ~ C;/llu - zhll v

vEV"

IIvllv=l

zllv

-liz -

zhllvl ~ C;/[CS1W(A)

-liZ -

Zhllv+-vl lIullvi

thus WheAl ~ (CECS)-lw(A) -liZ - Zhllv+-v JCE. From this follows the first part of (5) with C = (CECS)-l > 0 and .,,(h) = liZ - Zhllv+-vjCE, if lim sup II ZeAl - Zh(,x)lIv+-v h-tO>'EA

= O.

{11.2.6e)

The proof of (6e) is carried out indirectly. Its negation reads: there exist € > 0, ~ E A, h. - 0 with II Z(Ai) - Zh.(~)lIv+-v ~ € > O. Then there exist Ui E V with (11.2.6f) Due to (2b) and the compactness of A there exists a subsequence Aj E A, E V with limAj = A*, limuj = u* E V' in V'. (2c) and Theorem 8.2.2 show IIIZ(A*) - Zhi(A*)u*lIv - O. Together with (6c) we obtain

Uj

258

11 Eigenvalue Problems

II[Z(,xj) - Zhj(,xj)1ujllv ~ IHZ(,xj) - Z(,x*)]ujllv + II [Zhj (,xj) - Zhj(,x*)]ujllv + IIZ(,x*)[Uj - u*111v + IIZhj(,x*)[Uj - u*]/lv + II[Z(,x*) - Zhj(,x*)]u*IIv ~ 2GI,xj - ,x*1 + 2Gzlluj - u*IIv + II[Z(,x*) - Zhj(,x*)]U*/lv -+ 0 1

in contradiction to (6f). For the proof of the second part of (5) one replaces (6d) and the following estimate by Wh(,x)lluh/lv ~

lap.(uh - zh, v)1 ~ Gsllu h - zhll v

sup

for all uh E Vh,

1JEV" 1I1Jllv=l

laA(uh,v)1

sup 1JEV 1I1Jllv=l

=

sup 1JEV 1I1Jllv=l

lap.(uh - z,v)l;::: GE/llu h - zllv

for all u h E Vh.

;::: Gj;/[GS1Wh(,x) - IIZ(,x) - Zh(,x)llv+-vllluhllv

Let u E V with /lu/lv sup 1JEV 1I1Jllv=l

= 1 be selected such that

laA(u,v)1

=

inf

sup

laA(u,v)1

= w(,x).

uEV 1JEV lIullv=l 111Jllv=l

Since sup laA(u - uh, v)1 ~ Gsllu -uhllv, it follows for arbitrary u h E Vh that w(,x) ;::: GWh(,x) - IIZ(,x) - Zh(,x)IIv+-v - Gsllu - uhllv.

•

From (2c) and (6e) follows the second part of (5).

A corollary to Lemma 7 is Theorem B.2.B.1f Problem (B.1.1) for all f E V' is solvable, then ,x = 0 cannot be an eigenvalue, i.e., w(O) > O. Thus it follows that loN, ;::: WN,(O) ;::: lo := !Gw(,x) > 0 for sufficiently large i.

A second corollary concerns the convergence of the eigenvalues.

Theorem 11.2.8. Let (2a-c) hold. If ,xh, (i -+ 00, hi -+ 0) are discrete eigenvalues of (la) with ,xh, -+ Ao then Ao is an eigenvalue of (1.2a). PROOF. If Ao were not an eigenvalue then w(,x) ;::: 'f/O > 0 would be in the lo-neighbourhood of K£(Ao), since w(,x) is continuous (cf. Exercise 4(ii». There would exist ho > 0 such that "1(h) ~ G1}o/2 for all h ~ ho (G and "1(h) from (5». For all ,xh, E KE(,xo) with hi ~ ho the contradiction follows from (5): 0= Wh,(,xh.) ;::: GW(,xh. - "1(h i )

;:::

1 G'f/O - "2 G1}o

1

= "2G"10 > O.

•

Lemma 11.2.9. (Minimum Principle) Let (2a,b) hold. The functions w(,x) and Wh(,x) in the interior of A C () have no proper positive minimum.

11.2 Finite Element Discretisation

259

PROOF. Let L be the operator associated with a(·, .). Let >'*, with w(>.*) > 0, be an arbitrary point in the interior of A. For sufficiently small € > 0 we have Ke(>'*) C A and w(>.) > 0 in Ke(>'*). Thus (L - >.I)-1 is defined in Ke (>. *) and holomorphic. Cauchy's integral formula says (L - >'1)-1 =

~1 21n

( -

>.)-l(L - (I)-ld(

for all >. E Ke(>'*).

J8K..) ~ min{w«():( E 8Ke(>'*)} (cf. Exercise 4(i». Thus, w(>.) cannot assume a proper minimum in Ke(>'*). For Wh(>') the conclusion is the same .

•

The converse of Theorem 8 is contained in

Theorem 11.2.10. Let (2a-{!) hold. Let >.0 be the eigenvalue of (1.2a). Then there exist discrete eigenvalues >'h of (1 a) (for all h) such that limh-to >'h = >'0.

PROOF. Let € > 0 be arbitrary. According to Theorem 11.1.2, >'0 is an isolated eigenvalue: w(>.) > 0 for 0 < I>' - >.01 ~ € (€ sufficiently small). Since w(>.) is continuous and 8KE (>.0) is compact, we have that PE := min{w(>'): I>' - >.01 = €} is positive. Because of (5) and w(>'o) = 0 one obtains for sufficiently small h

for all >. E 8KE(>'0)' Thus Wh(>') must have a proper minimum in KE(>'O)' By Lemma 9 the minimal value is zero. Thus there exists a >'h E Ke(>'o) which is a discrete eigenvalue, Wh(>'h) = o. • The convergence of the eigenfunctions is obtained from Theorem 11.2.11. Let (2a-{!) hold. Let eh E Eh(>'h) be discrete eigenfunctions with lIeh llv = 1 and limh-to >'h = >'0. Then there exists a subsequence eh;. which converges in V to an eigenfunction e E E(>.o):

eE E(>.o), lIeh;. - ellv -

0

(i - (0),

lIellv =

1.

PROOF. The functions eh are uniformly bounded in V. Since V C L2(il) is compactly embedded (cf. (2b», there exists a subsequence eh ;. which converges in L2(il) to an e E L2(il): (11.2.7a)

260

11 Eigenvalue Problems

We define z = Z(Ao)e, zh. = Zh.(Ao)e according to (6a,b). According to Theorem 8.2.2 there exists an hl(E) > 0 such that

liz -

(11.2.7b)

zh'lIv ::5 E/2

for h. ::5 ht(f). The function e h • is a solution of

a,. (eh• , v) = (Ah. - JL) (eh• , t' )0

for all v E Vh•.

(11.2.7c)

A combination of eh• = Zh.(Ao)e (Le., (6b) for A = Ao) and (7c) yields

a,.(zh. - eh., v)

= F.(v) := (Ao - JL)(e _e h., v)o - (Ah. -

Ao)(eho, v)o (11.2.7d)

for all v E Vh.. Since IIF.II{" -+ 0 because Ah. -+ Ao and (7a), there exists an h2(f) > 0 such that IIFillv 1 ~ f/[2CE ] (CE from Lemma 6) and (11.2.7e) for hi ~ h2(f). (7b) and (7e) show that liz - eh'lIv ~ f for hi ~ min{ht(f), h2(f)}; thus liffii-+oo eh • = z in V. Therefore lime h• = z in L2(Q) c V also holds. (7a) proves z = e E V such that e = z = Z(Ao)e becomes aCe, v) = .\o(e, v)o. Therefore e = lime h• is an eigenfunction of (1.2a). In particular, lIellv = limllehollv = 1. • Exercise 11.2.12. Let (2a~) hold. Let Ah,eh,Ao, and e be as in Theorem 11. Show that (a) IfdimE(.\o) = 1 then also limh-+Odim Eh(Ah) = 1. (b) Let dimE(Ao) = 1. Then we have lime h = e in V for eh := eh/(e\e)v if I(eh,e)vl ~ 1/2 and eh := eh otherwise.

11.2.3 Quantitative Convergence Results The geometric and algebraic multiplicities of an eigenvalue Ao of (1.2a) agree if (11.2.8) dim Ker(L - .\01) = dim Ker(L - .\01)2. Lemma 11.2.13. Let (2a,b) and dimE(.\o) = 1 hold. Then (8) is equivalent to (e, e*)o t- 0 fOT 0 t- e E E(Ao), 0 t- e* E E*(Ao). PROOF. We have Ker(L -.\01) = E(.\o) = span {e}. dim Ker(L -AoI)2 > 1 holds if and only if there exists a solution v E V for (L - .\o1)v = e. According to Theorem 6.5.15c this equation has a solution if and only if (e, e*)o = O. • Let E(Ao) = span {e}, E*(Ao) and e* can be normalised so that

= span {e*}.

(e,e*)o

= 1.

Under the assumption (8) e (11.2.8')

11.2 Finite Element Discretisation

We define V:= {v e V: (v,e*)o = O}, V' = {v' e V': (v/,e*)o be the dual norm for 11·11" = 1I·lIv. For Problem (9): For

I e V',

find u

e

Vwith a),,(u,v)

= (f,v)o

for all v

261

= O}. Let 1I·lI v, e V,

(11.2.9)

one defines the variable corresponding to (3b)

w(>.):=

inf

la),,(u,v)l.

sup

(11.2.10)

uE" vE" lIullv=1 Ilvllv=l

Lemma 11.2.14. Let (2a,b), (8), and dimE(>.o) = 1 hold. Then there exists an € > 0 such that w(>.) ~ C > 0 lor alii>' - >.01 S €. Problem (9) has exactly one solution u e V with lIullv

s II/lIv'/w(>'),

il w(>.) > O.

t:

ce.

PROOF. Let V -+ V' be the operator associated with a(·, .): V x V -+ For 0 < I>' - >.01 € (€ sufficiently small) (L - >.1)u = I has a unique solution u e V. From I e V' follows

s

0= (f,e*)o = ([L - >.I]u,e*)o = (u, [L - >.I]*e*)o

= (Xo -

X)(u, e*)o,

i.e., u e V. Thus there exists (£ - >'1)-1: V' -+ V as the restriction of (L - >.1)-1 to V' C V'. For>. = >'0 Problem (9) has a unique solution according to Theorem 6.5.15c. Then there exists (£ - >'1)-1 for all >. e KE(>.o). According to Remark 5 (with V instead of V) , w(>.) must be positive in KE(>'O). The continuity of w(>.) proves w(>.) ~ C > O. In analogy to (4) one has lIullv = lIull" II/II",/w(>'). The bound by II/lIv'/w(>') results from

s

s

Exercise 11.2.15. Show that

II/lIv'

~ II/lIv' for all I

e V'.

Lemma 11.2.16. Let (2a-c), dimE(>'o) = 1, and condition (8) hold. Let >'h be discrete eigenvalues with limh--+O >'h = >'0. According to Exercise 12b there exists an eh e Eh(>'h) and e*h e Eh(>'h) with eh -+ e e E(>.o), e*h -+ e* e E* (>.0), (e, e*)o = 1. This enables one to construct the space Vh := {v h e Vh: (v h, eh)o = O} and the variable Wh(>'):=

inf

sup

uE "" Ilullv=1

vE "" II vllv=1

la),,(u, v)l.

Then there exists a C > 0 independent of hand>' e (() such that Wh(>') CWh(>')' For sufficiently small € > 0 and h, Wh(>') ~ 1J > 0 for all I>' ->'01 S PROOF. (a) First statement: there exists min{lIv + ae*hllv: a E

ce} ~ IIvllv/C

ho > 0 and C such that

or all v

e Vh

and all 0

< h ~ ho.

~ €.

262

11 Eigenvalue Problems

The proof is carried out indirectly. The negation reads: there exists a sequence Cti E 4J, Itt. -+ 0, Vi E Vh, with IIViliv = 1 and IIVi + aie*h'IIv -+ O. Thus there exist subsequences with Cti -+ a*, Vi -+ v* in L2(il). Evidently, Wi := Vi + aie*h, must have the limit w* = lim Wi = v* + a*e* in L2(il). Since lim IIWill£2(n) :::; CHm IIwiliv = 0, it follows that w* = 0, and thus v* = -a*e*. From 0 = lim (Vi, e*h,)o = (v*, e*)o = -a* IIe* 115 one infers a* = O. Thus the contradiction follows from 1 = lim IIviliv :::; limllwillv + limIiCtie*h'IIv = lim IIWiliv = O. (b) Second statement: Wh(A) ~ CWh(A) with C from (a). Because a>.(u,v) = a>. (1.£, V + ae*h) for 1.£ E Vh we have Wh(A)

= inf

sup

Ia>. (1.£, v) I/(IIuliv IIvllv)

O#uEV" O#tJEV"

~C

inf

sup maxla>.(u, v +ae*h)II[IIullvllv + ae*hllvl

O#UEV" O#tJEV" aEt:

= C inf

sup la>.(u, w)I/[IIullvllwllvl

~ C

sup

O#uE V" O#wE V"

inf

Ia>. (1.£, w)lIlIIullvllwllvl = CWh(A).

O#uE V" O#wE V"

(c) Let f > 0 be chosen such that Ao is the only eigenvalue in KE(Ao). For sufficiently small h, Ah is the only discrete eigenvalue in Ke(Ao). In the proof of Theorem 10 we have already used Wh(A) ~ rl' > 0 for A E 8K(AO), h:::; hO(f). From Part b follows that Wh(A) ~ 'TJ := 'TJ'C > 0 for A E 8KE(AO). According to Exercise 12a, a>. (u, v) = (I, v)o (v E Vh) is solvable for each f E Vh and all A E KE(AO) such that Wh(A) = 0 is excluded. Lemma 9 shows that Wh(A) ~ 'TJ > 0 in Ke(Ao). •

Exercise 11.2.17. Let (2a,b) hold. The functions 1.£, V E V satisfy (1.£, v)o =fi O. Let d(u, Vh) be defined as in (8.2.2). Show that if d(u, Vh) is sufficiently small then there exists a u h E Vh with

Lemma 11.2.18. Let (2a-c) hold. Let Ao be an eigenvalue with (8) and dimE(Ao) = 1. For sufficiently small h there exists eh E Eh(Ah) with

lie -

ehllv :::; C[lAo - Ahl + d(e, Vh)l.

PROOF. Let zh := Zh(Ao)e be the solution of (6b). Since e = Z(Ao)e, one has (11.2.11a) For all v E Vh we have

11.2 Finite Element Discretisation

263

al'(zh-eh, v) = (Ao - /l)(e,v)o - (Ah - /l)(eh,v)o = (Ao - Ah)(e, v)o + (Ah - /l)(zh - eh, v)o + (Ah - /l)(e - zh, v)o, SO that a>." (zh_e h, v) = (Ao - Ah)(e, v)o + (Ah - /l)(e - zh, v)o for all v E Vh. (11.2.11b) According to Lemma 16 we have I(e*,e*h)ol ~." > 0 for sufficiently small h. eh can be scaled so that (zh - eh , e*h)o = o. (llb) corresponds to Problem (9). Lemma 14 proves

Ilzh - ehllv ~ Wh(Ah)-lC[lAo - Ahl + lie - zhllv ) ~ C'[ lAO - Ahl + d(e, Vh)). Together with (lla) one obtains the statement.

•

Theorem 11.2.19. Let (2a---c) hold. Let AO be an eigenvalue with (8) and dimE(Ao) = 1. Let e E E(AO), e* E E*(Ao), llellv = 1, (e, e*)o = 1. Then there exist discrete eigenvalues Ah with (11.2.12) PROOF. Choose u h according to Exercise 17 such that lIe* - uhllv ~ 2d(e*, Vh), (e* - uh,e)o = o. Discrete eigenvalues Ah -+ AO exist by Theorem 10. From

o = a>.o (eh, e*) = a>." (e h , e*) -

(AO - Ah)(eh, e*)o =a>.,,(eh,e* _uh) -(Ao-Ah)(e\e*)o = a>." (e h - e, e* - uh) - (AO - Ah)( eh, e*)o = a>." (e h - e, e* - uh) - (AO - Ah)[(e, e*)o + (e h - e, e*)oJ

follows lAo - Ahl ~ C'lIe h - ellvllle* - uhllv + lAO - Ahll. By Lemma 18 there exists an eh E Eh (Ah) such that

lAO - Ahl ~ C'C/[lAo - Ahl + d(e, Vh)](IAo - Ahl + 2d(e*, Vh)]. From this one obtains (12) with C = 3C'C" for sufficiently small h, since lAo - Ahl-+ 0, d(e, Vh ) -+ 0, d(e*, Vh) -+ o. • Theorem 11.2.20. Under the assumptions of Theorem 19 there exist for e E E(AO), e* E E*(Ao) discrete eigenfunctions eh E Eh(Ah), e*h E Ei.(Ah) with (11.2.13)

PROOF. Insert (12) with d(e*, Vh ) estimate in (13) follows analogously.

~

const into Lemma 18. The second •

264

11 Eigenvalue Problems

In the following, let V C Hl(il). Theorem 11.1.5 proves E(AO) C H1+ S(il),

E*(>.o) C H1+ S(il).

(l1.2.14a)

for all u E E(>.o) U E*(AO).

(1l.2.14b)

Also, let (14b) hold (cf. (8.4.10)): d(u, V,,) ~ Ch8I1uIlHl+"(G)

Corollary 11.2.21. Let (2a), (2c), (14a,b) hold. Let Ao be the eigenvalue with (8) anddimE(Ao) = 1. Then there exists A",e" E E,,(A,,), e*" E Eh(A,,) such that

Occasionally eigenfunctions may have better regularity than is proven for ordinary boundary value problems. For example, let -Lle = Ae be in the rectangle il = (0,1) x (0,1) with e = 0 on First, Theorem 11.1.5 implies e E H2(il) n HJ(il), thus e E CO(il) (cf. Theorem 6.2.30). Thus e = 0 holds in the corners of il. According to Example 9.1.25 it follows that e E HS(il) for s < 4. As in Section 8.4.2 one obtains better error estimates for e - e" in the L2-norm. The proof is postponed until after Corollary 29.

r.

Theorem 11.2.22. Let (2a-c), (8), dimE(Ao) = 1, and (14a,b) with s = 1 hold. Let a(·,·) and a*C·) be H2- regular, i.e., for f E L2(il) , a~(u,v) = (I,v)o and a~(v,u*) = (v,J)o (v E V", JL from Lemma 6) have solutions u, u* E H2(il). Let e E E(Ao) and e* E E*(Ao). Then there exist A", e" E E,,(A,,), and e*" E Eia(A,,) with

lie - e"IIL2(G) ~ C'h2, If(14a,b) also hold with some s

> 1,

lIe* - e*"IIL2(G) ~ C'h2. one must replace C'h2 by C'h1+B.

11.2.4 Complementary Problems In Problem (9) we have already encountered a singular equation which nevertheless was solvable. In the following let >.0 be the only eigenvalue in the disc Kr (Ao). The equation a). (u, v) = (I, v)o

for all v E V

(11.2.16a)

is singular for A = Ao. For A ~ >.0 Equation (16a) is ill-conditioned. In the following we are going to show that Equation (16a) is well-defined and wellconditioned if the right-hand side f lies in the orthogonal complement of E*(Ao):

11.2 Finite Element Discretisation

f 1- E*(AO)

,i.e., (f,e*)o = 0 for all e* E E*(AO».

265

(l1.2.16b)

In the case of A = AO , with u, u + e (e E E(AO)) is also the solution. The uniqueness of the solution is obtained under the conditions (8) and (16c): (l1.2.16c)

u 1. E*(>.o).

Remark 11.2.23. Let (2a,b) and (8) hold. Let >.0 be the only eigenvalue in Kr(AO). Then (16a,b) has exactly one solution u for all IA - Aol ~ r which satisfies (16c). There exists a C independent of f and A such that lIuliv ~

Cllfllv

1 •

PROOF. This follows from Lemma 14 in which the assumption E(AO) = 1 is not necessary. • The finite element discretisation of Equation (16a) reads: Find u h E Vh with a>. (u h , v) = (f' v)o

for all v E Vh.

(11.2.17)

In general, Equation (17) need not be well-defined, even assuming (16b). For the sake of simplicity we limit ourselves in the following to simple eigenValues: dimE(>.o) = 1. Equation (17) is replaced by (18a): Find u h E Vh

with a>.(uh,v)

= (f(h),v)O for all v E Vh

(11.2.18a)

with f(h) 1. Ei.(Ah),

(1l.2.18b)

uh 1. Ei,,(Ah).

(l1.2.1Sc)

Exercise 11.2.24. Show that pSa-c) ~s equivalent to: Find u h E Vh with a>.(u\ v) = (f(h) , v)o for all v E Vh with Vh = Vh n Ei,,(Ah).i as in Lemma 16. Lemma 16 proves the Remark 11.2.25. Let (2a-c), (S), dim E(AO) = 1 hold. Let AO be the only eigenvalue in Kr(AO). Then there exists an ho > 0 such that, for all h ~ ho and all A E Kr(AO), the Problem (lSa,b) has a unique solution u h = Uh(A) which satisfies the additional conditions (lSc). Further there exists a C independent of h, A, and f(h) such that lIuhllv ~ Cllf(h)lIv. If Ei.(Ah) =1= E*(AO), f from (16b) need not satisfy condition (lSb). If eh E Eh(Ah) and e*h E Ei.(Ah) with (e h, e*h)o = 1 are known, one can define

f(h) := Qhf := f - (f' e*h)oe h . f(h) satisfies (lSb) since Qh represents the projection on E;;(Ah).i.

(11.2.19)

266

11 Eigenvalue Problems

Exercise 11.2.26. Let u.l E(Ao), dimE(>.o) (eh,e*h)o = 1. Show that

= dim Ell. (>'11.) = 1, Ilehl/v = 1,

d(u, Eh(>'h)l. n VII.) := inf{IIu - vhl/v, vII.

E

VI., vII. .l Ei,.(>'h)}

S C[d(u, VII.) + l/ulivinf{l/e* - e*hllvl:e*

E

E*(Ao)}].

Theorem 11.2.27. Let (2a-e), (8), dimE(>.o) = dim Eh(>'h) = 1 hold. Let Ao be the only eigenvalue in Kr(Ao}. Let h be sufficiently small such that Uollowing Remark 25) the Problem (18a-e) is solvable. For the solutions u and u h 01 (16a-e) and (18a-e) the error estimate

I/u-uhllv S C[d(u, Vh)+I//IIvdnf{l/e*-e*hl/vl:e*

E

E*(Ao)}+II/(h)-/l/v/] (11.2.20)

holds, with C independent 01 I, 1(11.), and h. PROOF. Repeat the proof of Theorem 8.2.1 for a>.(-,·) instead of a(·, .). Here one must choose w E VII. with w.l Eh(>'h). Furthermore, (8.2.3) becomes

a>.(uh - u, v) = (f(h) - I, v)o for all v EVil.. EN agrees with Wh(>') ~ "1 > 0 (>. E Kr(Ao))) (cf. Lemma 16). I/u - wllv is estimated with the aid of Exercise 26, with IIuliv S II/IIvl being added. •

Corollary 11.2.28. 111(11.) is defined by (19), then inequality (20) becomes IIu - uhllv S C'[d(u, VII.) + I//livl inf{lle* - e*hl/v: e* E E*(Ao)}]. (12.2.21a)

PROOF.

li h) - Ilivl S CI(f, e*h)ol = CI(f, e* - e*h)ol S Cl//livll/e* - e*h IIv.

•

Corollary 11.2.29. II additionally the assumptions u E H1+ a(n), (14a), and d(u, VII.) S Ch 8 1UI1+a hold lor u E H1+ S (n) then (21a) yields the estimate (1l.2.21b) It remains to add the PROOF. (Theorem 22) For e E E(Ao) there exists ell. E Eh(>'h) with I := e - eh .l E(>.o) and 1/11 = I//I/v S Cha = Ch. According to Remark 25 the problem a>.o (v, w) = (v, 1)0 has a solution w .l E* (>'0) for all v E V. The assumption of regularity yields w E H2(n), Iwl2 S Cillo such that wh E VII. exists with w h .l Ei,.(>'h), Iw - whit S Chlwl2 S C'hi/io. The value

11.3 Discretisation by Difference Methods

267

a>"o(f,w h ) = a>"o(e,w h ) -a>"o(eh,wh ) = 0 -a>"o(eh,wh ) = (-'0 - Ah)(eh, wh)o - a>"h (eh, w h) = (-'0 - Ah)(eh, wh)o can be bounded by Ch21whlolehlo ~ C'h 2 1flo (cf. (15)). From Ifl~ = (f,f)o = a>"o(f,w) = a>"o(f,w - w h ) + (AO - Ah)(e\wh)o ~ C[C'hlfltlflo + h21flol

and Ifb ~ Ch one infers Iflo ~ C'h2 • The same method is then applied to le* - e*hlo. • Exercise 11.2.30. Formulate the conditions for and the proof of the error estimate lIu-UhllL2(s) ~ Ch21ul2 (u, u h from Corollary 28). Hint: Decompose u - u h in 11 + 12 with 11 with 11 .1 E*(-'o) and 12 E E(AO).

11.3 Discretisation by Difference Methods In the following we limit ourselves to the case of a difference operator of the order 2m = 2. The differential equation Lu = f with homogeneous Dirichlet boundary condition is replaced, as in Sections 4 and 5, by the difference equation LhUh = fh. The eigenvalue equations Le = Ae, L*e* = >':e* are discretised by (11.3.1)

Lh is the transposed and complex-conjugate matrix to Lh . The general assumptions of the following analysis are:

v = HJ(n), n E co· 1 bounded,

(11.3.2a)

a( u, v) = (Lu, v)o is HJ (n)-coercive,

(1l.3.2b)

Consistency condition ILhRh -

RhLI-h-2 ~ Ch.

(11.3.2c)

Condition (2c) has been discussed in Section 9.2.2. Assume furthermore that Lh is Hk-coercive. For suitable J-t E m

L",.h := Lh - J-tI (I: identity matrix) is thus Hk-regular: (1l.3.2d) Furthermore, let

L", := L - J-tI H 2(n)-regular,

(11.3.2e)

Le., IL;112+-0 ~ C. The boundedness of Land Lh reads ILI-1+-1 ~ C,

IL h l-1 0 one has to show I(Rh - Rh)ul Sf for h S h(f). Since HJ(J2) is dense in L2(J2) there exists au E HJ(J2) with Iu-ulo S f/[2IRh -Rhlo+-o) such that I(Rh -Rh)(U-u)lo S f/2. By (2i) there follows I(Rh - Rh)ulo Chlu!t f/2 for h S h(f) := f/[2C!u!t). Altogether this gives I(Rh - Rh}:.£lo f. (b) From (2k) one infers 0 = limh-+O(Rhu, Ph'u)o = limh-+o(PhRhu, u)o = (u,u)o thus u = o. •

s

s

s

Exercise 11.3.3. Let Ah := I - 8;;8; - 8:8; and A:= 1- .1. Show that

11.3 Discretisation by Difference Methods

lul~

269

IUhl~ = (AhUh, Uh)O, IL~ul~l = (v,L~u)o for v = A-IL~u E HJ(n),

= (Au, u)o,

IL~,hUhl~l = (Vh, L~,hUh)O for Vh = Ah"l L~,hUh' lim I(AhRh - RhA)ul-l = 0 for all U E HJ(n).

h->O

The following analysis is tailored to the properties of the difference methods under discussion. We have tried to avoid a more abstract theory that is applicable to finite elements as well as difference methods. This kind of approach can be found in Stwnmel [I] and Chatelin [I]. The variable Wh(A) is now defined by

Wh(A):=

sup I(L~,hUh' vh)ol =

inf

IUh!1=llvhll=l

inf

IL~.hUhl-l.

(11.3.3)

IUh!1=l

As in Exercise 11.2.4 we have Wh(A) = 0, if A is an eigenvalue of L h; Wh(A) = 1/ILi".ilt+--l otherwise.

(11.3.3')

The analogue of Lemma 11.2.7 reads Lemma 11.3.4. Let A C C[) be compact. Let (2a-m) hold. Then there exist variables C > 0 and '11(h) - 0 (h - 0), independent of A E A, such that

Wh(A)

~

CW(A) - '11(h),

W(A) ~ CWh(A) - '11(h)

for all A E A, h > o. (11.3.4)

PROOF. (a) Since A is compact, W(A) is continuous, and Wh(A) is equicontinuous in A, it is sufficient to show that limh->owh(A) ~ CW(A), and W(A) ~ Climh->owh(A) for all A E A with C > o. (b) Define for A E A and Uh with IUhh = 1 and IL~,hUhl-l = Wh«A)

u:= PhUh,

Zh:= (A - JL)L;,iuh,

Z = (A - JL)L;lu

with JL from (2e). Without loss of generality it can be assumed that J.t We have

¢ A.

zit = 11\(Uh - Zh) + PhZh - zit ~ IPhlt+-lluh - Zh!t + IPhZh - zit, IPhZh - zit = IPh[Zh - RhZ) - [I - PhRh]zlt

Iu -

~ IPhh+-llA - JLIIL;.i(.UhL,. - L,.,h R h]L;lu + L;,i[1 - '&"Ph]Uhh

+ 1(1 -

PhRh)zlt - 0 (h - 0)

(cf. (2gj,m» so that (11.3.5a)

270

11 Eigenvalue Problems

As in (2.6d) one obtains Iu - zit ~ C 2W(A)lUIt,

C 2 > o.

(11.3.5b)

From

Wh(A) = IL,x,hUhl-l = ILJ',hUh + (JL - A)Uhl-l ~ (LJ',hUh + (JL - A)Uh' Uh)O ~ -IJL - Alluhl~ + (LI-',hUh, Uh)O ~ -CAluhl~ + CEluhl~ = -CAluhl~ + CE with CA := max{IJL - AI: A E A} > 0 follows

IUhl~ ~ CAl[CE -Wh(A»). Either Wh(A) ~ CE/2 from which the statement results directly, or Wh(A) $ C E /2 which yields (11.3.5c) We want to show that there exists ho > 0 and Cp = Cp(Co) such that for all Uh with IUhlo ~ COIUhh and h $ ho.

IUhh $ CplPhuhh

(11.3.5d)

The negation of (5d) reads: there exists Uh with IUhh = 1, IUhlo ~ Co and IPhUhh -- 0 (h -- 0). From IRhPhUhlo $ IRhPhUhh $ CIPhUhh -- 0 and IUh -RhPhUhlo ~ II -RhPhlo~lluhh ~ Ch -- 0 follows IUhlo -- 0 in contrast to IUhlo ~ Co. Thus we have (5d). Together with LJ',h(Uh - Zh) = L,x,hUh, (2d) and (5a,b,d) yield the first of the inequalities (4):

Wh(A) = IL,x,hUhl-l = ILJ',h(Uh - zh)l-l ~ CEluh - Zhlt ~ CECllu - zit - 0(1) ~ C EC l C2W(A)lult - 0(1) ~ CW(A) - 0(1) with C:= CEClC2/Cp > O. (c) Let

W(A)

~

> 0 be arbitrary; U E HJ(.f1) with Iuh = 1 can be selected such that IL,xul-l - f. Set Uh := Rhu. According to Exercise 3 it holds that

f

IL,xul~l

= (v, L,xu)o for v = A- l L,xu E HJ(.f1),

IL,x,hUhl~l = (Vh' L,x,hUh)O

A = 1-.1,

for Vh:= Ah l L,x,hUh.

From

Rhv - Vh

= Ahl[AhRh -

RhL,xU - L,x,hUh

RhAjA- l L,xu + Ah1[RhL,x - L,x,hRhju -- 0

= [RhL,x - L,x,hRhju -- 0,

(v, L,xu)o - (RhV, RhL,xU)O = ([I - RhRhjv, L,xu)o -- 0, for h -- 0 (cf. (2m), (2k», one infers IL,x,hUhl-l -- IL,xul-l and

W(A) ~ IL,xul-l -

f

~ IL,x,hUhl-l -

f -

0(1) ~ Wh(A) -

f -

0(1)

(h -- 0)

11.3 Discretisation by Difference Methods

for each

€

> O. Thus w{>.)

~

limh ......owh{>.) has been proved.

271

•

Corollary 11.3.5. Under the assumptions of Lemma 4 the following holds: If there exists L-;:1 for all >. E A then there exists an ho > 0 such that L>..h for all >. E A and h :s he is H~-regular: sup{IL-;:~hi--1: >. E A, h ::; he} ::; c .

.

PROOF. We have assumed w{>.) > 0 in A so max{w{>.): >. E A} =: '" > O. Choose he according to Lemma 4 such that Wh(>') ~ Cw{>.) - !C'" ~ !C'" for h::; ho. Then IL.\~h-+-l ::; 2/(C",) for all >. E A, h ::; he. •

.

The proof of Theorem 11.2.8 can be carried over without change and results in

Theorem 11.3.6. Assume (2a-m). If >'h. (hi -+ 0) are discrete eigenvalues of Problem (1) with >'h, -+ >.0 then >'0 is an eigenvalue of (1.2a). Lemma 11.2.9 and Theorem 11.2.10 can also be carried over without changes.

Theorem 11.3.7. Let (2a-m) hold. Let >.0 be an eigenvalue of (1.2a). Then there exist discrete eigenvalues >'h of (1) Vor all hI such that limh-+o >'h = >'0. Theorem 11.3.8. Let (2a-m) hold. Let eh be discrete eigenfunctions with lehh = 1 for >'h, where >'h -+ >.0 (h -+ 0). Then there exists a subsequence eh. such that Phoeh. converges in HJ{n) to an eigenfunction 0 ~ e E E(>.o). Purther, we have leh, - Rh,elt -+ o.

PROOF. Because IPhehh :s C (cf. (2g) the functions eh := Pheh are unifomIly bounded. HJ(n) is compactly embedded in L2(n) (cf. (2a) and Theorem 6.4.8a) such that a subsequence e h, converges in L2(n) to an e E L2{n): leh • - elo -+ O. We have in particular IRhe - ehlo ::; IRh(e - eh ) - (RhPh - I)ehlo

-+

0 (h = hi

-+

0

-+

0).

Estimate (2c) yields IRh Z - zhlo ::; IRhZ - zhh ::; Chlelo

for Z = {>'o - J.L)L;;.le,

Zh:=

(>.0 - J.L)L;,~Rhe.

From L J4 •h(Zh - eh) = (>'0 - J.L)(Rhe - eh) + (>'h - >.o)eh -+ 0 in H;;l follows IZh - ehl! -+ 0 such that IRh(e - z}lo -+ 0 (h = ht -+ 0) results. Lemma 2b shows that e = Z E HJ(n), and 0 = e E E(>.o) is excluded because of Ilim Rh,eh = llimehh = 1. •

Theorem 11.3.9.

Let (2a-m) hold. Let e h be the solution of the discrete eigenvalue problem Lheh = 'Xhe h with lehlt = 1, limh......O >'h = >.0. Then there

11 Eigenvalue Problems

272

exists a subsequence eh. such that Ph.et in HJ(.!1) converges to an eigenfunction 0 # e* E E*(AO)' Furthermore, leh. - Rh.e*h -+ O. Proof Sketch. The proof is not analogous to that of Theorem 8 since the consistency condition (2a,m) does not necessarily imply the corresponding statements for the adjoint operators. One has to carry out the following steps: (a) e*h := Pheh -+ e* converges in L2(.!1) for a subsequence h = kt -+ O. (b) For Z = (AO - p.)L~-le* and Zh := (Ao - P.)L~:hleh the following holds:

Z - RhZh

= (Ao -

p.)L~-l[(e* - R;;'e;;') + (R;;'L~.h - L,.Rh)]L~:hle;;'.

For each v E L2(.!1) we obtain

(v, Z - R;;'Zh)O

= (Ao -1'){(L;lv, e* - e*h)o + ([P'; - Rh]L;lv, eh)o

+(L;.k[L,..hRh - RhL"lL;lv, eh)o} -+ 0 for h = kt -+ 0 (cf. (2i), (2c». (c) IZh - ehh -+ 0 may be inferred from L~.h(Zh - eh) = (AO - Ah)eh -+ O. In particular, (v, RhZh - Rhe;;')o -+ 0 for each v E £2(n). (d) (v,R;;'e;;' - Rheh)O = ([Rh - Rhlv,eh)o -+ 0 for each v E L2(.!1) (cf. (21». (e) (v, Rheh - e*)o -+ 0 for each h = hi -+ O. (f) From (b) and (e) follows (v, Z - e*)o = 0, thus Z = e* E E*(Ao), where e* # o. Exercise 11.3.10. Carry over Exercise 11.2.12 to the present situation. In Lemma 11.2.16 we defined Wh(A). Now set

Vh := {Uh: (uh,eh)h = O} = {eUl., L;;'eh = Aheh' Ah W(A) := inf sup I(L~.hUh' vh)ol/(Iuhh IVhh).

-+

Ao, (11.3.6)

O#U"EV" O#v"EV"

A basic condition for the following is dim Eh(Ah) = 1,

Eh(Ah) = span {eh},

E;;'(Ah) = span {eU.

(11.3.7)

Here

E(Ah) := {Uh E H~: LhUh = AhUh},

E;;'(Ah):= {Uh E H~: LhUh = XhUh}.

Exercise 10 shows that (7) is valid for h

~

ho if dimE(Ao)

= 1.

Lemma 11.3.11. Let (2a-m), dimE(Ao) = 1, and (2.8) hold. Then there exist ho > 0 and a C > 0 that is independent of h ~ ho and A E 4J such that Wh(A) ~ CWh(A) for h ~ ho. For sufficiently small f > 0 and h, Wh(A) ~ 'TJ > 0 for alllA - Aol ~ f. PROOF. (a) There exists a C > 0 such that

11.3 Discretisation by Difference Methods

273

Indirect Proof: Assume that there exists a sequence Vh. with hi --+ 0, IVh.h = 1, ai e CU, Who := Vh. + aie'h., IWh.ll --+ O. For a subsequence of {hd

ai

--+

a*,

Ph.Vh.

--+

v*

and Ph.et

--+

e* -lOin L2(fJ),

converge. From 0 = (vh.,e'h.)o = (vh.[I - Ph'.Ph.]e;;')o + (Ph.Vh.,Ph.e'h.)o--+ (v*,e*)o one infers (v*,e*)o' = 0, a* = (w*,~*)o/(~*,e*)o = 0, V* = 0'. The contradiction results from 1 = lim IVh. It = lim IWh. It = o. (b) The rest of the proof runs as in Lemma 11.2.16. • Lemma 11.3.12. Let (2a-m), dimE(Ao) = 1, and (2.8) hold. One may choose 0 -I e e E(-\o) and eh e Eh(Ah) so that IRhe - ehlt $ C[h+ 1-\0 - Ahll· PROOF. We have that e = (-\o-IJ)L;le e H2(n)nHJ(fJ). Assume also that Zh := (-\0 - JL)L;,lRhe. The inequality (9.2.22) implies IRhe - zhlt $ Chlel2. For sufficiently small h we have (eh, e'h) -10 so that one can scale eh such that (eh - zh,e'h)o = O. From

L:>.",h(eh - Zh) = (Ah - Ao)Rhe + (Ah - JL)(Zh - Rhe) = (Ah - -\o)Rhe + (Ah - JL)[(Zh - Rhe) + (Rh - Rh)ej one infers that leh - zhh $ CIIAh - -\01 + h] (cf. Lemma 11) since I(Rh Rh)el-l = O(h). The statement follows from this. • Lemma 11.3.13. Under the assumptions of Lemma 12 we have the inequality IAh - -\01 $ Ch. PROOF. Choose eh, eh such that (Rhe - eh, eh)o = (eh, Rhe* - eh)o = O. For the Rayleigh quotient Xh := (LhRhe, Rhe*)o/(Rhe, Rhe*)O we then have

I~h - Ahl $ C1Rhe - ehltlRhe* - ehll $ €h[h + lAO - Ahll with €h := CIRhe* - ehlt· From (2c, j) one infers that

(LhRhe, Rhe*)O - (Le, e*)o = ([LhRh - RhLje, Rhe*)O + (Le, [R'hRh - Ije*)o = O(h)

and I~h - Aol = O(h) such that IAh - Aol $ Ch + €h\Ah - -\0\. For sufficiently small h we have €h $ (cf. Theorem 9, Exercise 10) thus \Ah -Aol $ 2Ch. •

!

Lemmata 12 and 13 give

274

11 Eigenvalue Problems

Theorem 11.3.14. Let (2a-m), E(AO) = span {e}, and (2.8) hold. There exists eh E Eh(Ah) with IRhe - ehh S Ch. Since IRhe* - ehh = 0(1) or even IRhe* - ehh = O(h), according to Theorem 11.2.19 one expects that lAo - Ahl = o(h) [resp. O(h2)). In general, this estimate is false as the following counterexample shows. Example 11.3.15. The eigenvalue problem -u" + u' = AU in (0,1) with u(O) = u(l) = 0 has the solution u(x) = exp(Ax)sin(1rx) with A = 1/2. The eigenvalue is Ao = 1r2 + 1/4. One calculates that the discretisation a+ u = AU has the eigenva.lue Ah = h- 2[2 - cos(1rh)(eA'h

+ e-A'h) -

-a-a+u+

isin(1rh)(e A'h - e-A'h))

+ h-l[cos(1rh)eA'h -1 + isin(1rh)e A'h] = 1r2 + ~ + (~ _ 3;2) h + O(h2) with A' := 10g(1 such tha.t

lAo - Ahl turns out no better than O(h).

h)/(2h)

12 Stokes Equations

12.1 Systems of Elliptic Differential Equations In Example 1.1.11 we have already stated the Stokes equations for - LlUl + 8p/8x l = ft, - LlU2 + 8p/8x2 = 12, - 8ul/8xl - 8u2/8x2 = o.

n ern?: (12.1.1ad (12.1.1a2) (12.1.1b)

In the case of n c rn.s another equation -Llus + 8p/8xs = is needs to be added, and the left-hand side of (1b) must be supplemented by -8us/8xs. A representation independent of the dimension can be obtained if one takes together (U1' U2) [resp. (Ul, U2, us)] as a vector u satisfying, in n, the equations -Llu+Vp=f -divu= O.

(12.1.2a) (12.1.2b)

Here, div is the divergence operator n

divu = 'L8ud8x., i=l

and n is both the dimension of n c mn and the number of components of u(x) E rn.n . Only n S 3 is of physical interest here. In fluid mechanics the Stokes equation describes the flow of an incompressible medium (neglecting the inertial terms) and describes the velocity field. With x E rn.n , Ui(X) is the velocity of the medium in the Xi direction; the function p denotes the pressure. Up to now we have not formulated any boundary conditions. In the following we limit ourselves to Dirichlet boundary values: u=O

onr.

(12.1.3)

This implies that the flow vanishes at the boundary. No boundary condition is given for p. Since both the pairs (u,p) and (u,p+const) satisfy the Stokes equations (2a,b), (3), one obtains: Remark 12.1.1. p is determined only up to a constant by the Stokes equation (2a,b) and the boundary conditions (3).

W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, DOI 10.1007/978-3-642-11490-8_12, © Springer-Verlag Berlin Heidelberg 2010

275

276

12 Stokes Equations

The Stokes equations have been chosen as an example of a system of differential equations. It remains to investigate whether Equation (2a,b) is elliptic in a sense yet to be defined. Even though the functions Ui, for given p, are solutions of the elliptic Poisson equations -LlUi = Ii - Op/OXi, in determining p, (2a,b) do not provide an elliptic equation, in any current sense of elliptic. A general system of q differential equations for q functions U l , " ' , u q can be written in the form q

LLijuj

= Ii

in

n c IR7l

(1 ~ i ~ q)

(12.1.4a)

j=1

with the differential operators

Lij =

L

caDa

(1 ~ i,j ~ q).

(12.1.4b)

lal:5k.; The equations (4a) are summarised as Lu = I where L is the matrix (Lij) of differential operators and U = (Ul,'" ,uq)T, I = (ft,···, Iq)T. The order of the operator Lij is at most ~j. Let the numbers ml. ... , mq, mL ... , m~ be chosen so that (12.1.5) ~j ~ ffii + mj (1 ~ i,j ~ q). As the principal part of Lij one defines

L~:=

L

caDa.

lal=m.+mi

< mi + mj, Ca = 0 holds for lal = mi + mj, and thus L~ characteristic polynomial associated with L~ reads:

If ~j

= O. The

and forms the matrix function

Definition 12.1.2. (Agmon-Douglis-Nirenberg [1]) Let (5) hold for mj. The differential operator L = (Lij) is said to be elliptic in x En if

ffii,

(12.1.6a) L is said to be uniformly elliptic in

n if there exists an

f

> 0 such that (12.1.6b)

with 2m := 2:;=1 (mi +mi). To be more precise one should call L elliptic with indices mi, mj, since the definition does depend on mi, mj. In connection with

12.1 Systems of Elliptic Differential Equations

277

this problem, as well as for an additional condition for x E r, q = 2, see the original paper of Agmon-Douglas-Nirenberg [1]. Further information on this subject can be found in Cosner [1].

Exercise 12.1.3. (a) Show that the numbers mi, mj are not unique. If mi and mj satisfy the inequality (5), then so do ffii -k and mj+k. The definition of L~ is independent of k. (b) Show that for q = 1, Le., for the case of a single equation one recovers from (6a) the Definitions 1.2.3a [resp. (5.1.3a)]; (6b) corresponds to (5.1.3a'). (c) For a first-order system (Le., kij = 1, mi = 1, mj = 0), (6a) coincides with Definition 1.3.2. In order to describe the Stokes equations in the form (4a, b) we set q:=n+l,

Lii = -..:1,

u=(Ul, ... ,un,p)T,

Liq = -Lqi = O/OXi for 1

The orders are numbers

~i =

2,

~q

= k qi = 1 (i

~

f=(ft"",fn,O?, ~

i ~ n,

Lij = 0 otherwise.

n), and k ij = 0 otherwise. The

mi = m~ = 1 for 1 ~ i ~ n = q - 1,

mq = m~ = 0

satisfy (5). L~ coincides with Lij and is independent of x: L{;(~) = _1~12,

L:;(~) = -L~(~)

= ~i

for i ~ n,

Lf;(~)

=0

otherwise.

From this we see

IdetLP(~)1 = /det

[

o

_1~12

0

-6 so that (6b) is satisfied, with

co

= 1.

Exercise 12.1.4. In elasticity theory the so-called displacement function n c rn.3 _ rn.3 is described by the Lame system:

u:

J.L..:1u + (,\ + J.L)Vdivu = f

in il,

U

= cp

on

r

(ft, >. > 0). Show that this system of three equations is uniformly elliptic: Idet L P (~)I = J.L2 (2J.L + '\)1~16.

For the treatment of Stokes equations we will use a variational formulation in the next section. For reasons of completeness we point out the following transformation.

Remark 12.1.5. Let n = 2 and thus U = (UbU2). Because divu = 0 there exists a so-called stream function i.P with Ul = oi.P/OX2, U2 = -0iP/OX1.

12 Stokes Equations

278

Insertion in Equation (1al,2) results in the biharmonic equation Ll2cp = a/2/aXl - afdaX2. The boundary condition (3) means VCP = 0 on r. This is equivalent to acp/an = 0 and acp/at = 0 on r where a/at is the tangential derivative. acp/at = 0 implies cp = const on r. Since the constant may be chosen arbitrarily, one sets cp = acp/ an = 0 on r.

12.2 Variational Formulation 12.2.1 Weak Formulation of the Stokes Equations Since U = (Ub···, un) is a vector-valued function, we introduce HA(n) := HJ(,G) x HJ(n) x ... x HJ(J])

(n-fold product).

A corresponding definition holds for H-l(n), H2(n), etc. The norm associated with H6(n) will again be denoted by I· It in the following. According to Remark 12.1.1 the pressure component p of the Stokes problem is not uniquely determined. In order to determine uniquely the constant in p = V+const, we standardise p by the requirement p dx = o. That is the reason why in the following p will always belong to the subspace L~(n) C L2(n):

In

L~(n) := {p E L2(n):

L

p(x) dx = O}.

To derive the weak formulation we proceed as in Section 7.1 and assume that and p are classical solutions of the Stokes problem (1.2a,b). Multiplication of the ith equation in (1.2a) with Vi E c8"(n) and subsequent integration implies that U

In =

l

fi (X)Vi (x) d(x) = LI-Llui(x) + ap/aXi]Vi(X) dx

(12.2.1a)

{(VUi(X), VVi(X)} - p(X)aVi(X)/aXi} dx for Vi E coc(n), 1:CS; i

:cs;

n.

Summation over i now gives

l

{(Vu(x), Vv(x)) - p(x) div, vex)} dx =

l

(f(x), vex)) dx,

(12.2.1a')

where the abbreviation

is used. Equation (1.2b) is then multiplied with some q E L~(n) and integrated, giving

12.2 Variational Formulation

-In

q(X) divu(x) dx = 0 for all q E

L~(il).

279

(12.2.1b)

With the bilinear fonus a( u, v) :=

In

(Vu(x), Vv(x)} dx for u, v E H6( il),

b(P,v) := - iP(X)diVV(X)dx for P E

L~(il),v E HA(il),

(12.2.2a) (12.2.2b)

we obtain the IVweak formulation of the Stokes problem as (3a-c): Find

U

a(u,v)

E HA(il)

and P E L~(il)

+ b(P,v) = f(v):=

b(q, u)

=0

In

such that

(12.2.3a)

(f,v)dx

for all v E H6(fl), (12.2.3b)

for all q E L~(il).

(12.2.3c)

In (3b) we first replace "v E H6(il)" by "v E G~(il)". Since both sides of (3b) depend continuously on v E H6(il) and since G~(il) is dense in H6(il), (3b) follows for all v E HJ(il). Remark 12.2.1. A classical solution u E G2(il)nHfi(il), P E Gl(il)nL~(il) of the Stokes problem (1.2a,b), (1.3) is also a weak solution, i.e., a solution of (3a-c). If conversely (3a-c) has a solution with u E G 2 (!}), P E Gl(!}), then it is also the classical solution of the boundary value problem (1.2a,b), (1.3). PROOF. (a) The above considerations prove the first part. (b) Equation (3c) implies divu = O. Let i E {I,··., n}. In Equation (3b) one can choose v with Vi E G8"(fl), Vj = 0 for j ¥= i. Integration by parts recovers (la) and hence the ith equation in (1.2a). •

12.2.2 Saddlepoint Problems The situation in (3a-c) is a special case of the following problem. We replace the spaces H6(il) and L~(il) in (3a-c) by two Hilbert spaces V and W. Let a(·, .): V x V b(., .): W x V -

m. m.

a continuous bilinear form on V x V,

(12.2.4a)

a continuous bilinear form on W x V,

(12.2.4b)

11 E V',

hE W'.

(12.2.4c)

In generalisation of (6.5.1) we call b(., .): W x V if there exists a Gb E m. such that

m. continuous (or bounded),

Ib(w,v)1 ~ Gbllwllwllvllv for all w E W;v E V. The objective of this chapter is to solve the problem (5a-c):

280

12 Stokes Equations Find v E V

and w E W

with

(12.2.5a) (12.2.5b) (12.2.5c)

a(v,x) + b(w,x) = h(x) for all x E V; b(y,v) =h(Y) forallyEW. Formally, (5a-c) can be transformed to the form Find u E X

with c(u, z)

= f(z)

for all z EX

(12.2.6a)

if one sets: X

=V

x W;

c(u, z) := a(v, x) + b(w, x) + bey, v)

f(z)=/t(x)+h(Y)

and

for u= (:).z= (:).

(12.2.6b)

Exercise 12.2.2. Show that (a) c(·, .): X x X -+ lR is a continuous bilinear form. (b) Problems (5a-c) and (6a,b) are equivalent. That the variational problems (5) and (6) must be handled differently than in Section 7 is made clear by the following remark. Remark 12.2.S. The bilinear form c(·,·) in (6b) cannot be X-elliptic. PROOF. We have c(u, u)

= 0 for all u = (~).

•

In analogy to (6.5.9) we set

J(v, w) := a(v, v) + 2b(w, v) - 2h(v) - 2h(w), and therefore J(v, w) = c(u, u) - 2f(u) for u = ( : ) . For v E V, w E W we know J( v, w) is neither bounded below nor above. Therefore the solution v*,w* of (5a-c) does not give a minimum of J; however, under suitable conditions, v·,w* may be a saddle point.

Theorem 12.2.4. Let (4a-c) hold. Let a(-,·) be symmetric and V-elliptic. The pair v* E V, w* E W is a solution of the problem (5a-c) if and only if J(v*,w):5 J(v*,w*):5 J(v,w*)

for all v E V;w E W.

(12.2.7)

Another equivalent characterisation is J(v*,w*)

= minJ(v,w*) = wEW max min J(v, w). tJEV tJEV

PROOF. (aa) Let v*,w* solve (5). The expression in brackets in

(12.2.8)

12.2 Variational Formulation

281

J(v, w*) - J(v*, w*) = a(v* - v, v* - v) -2[a(v*, v* - v)

+ b(w*, v* -

v) - ft(v* - v)]

vanishes because of (5b). Since a(v* - v, v* - v) > 0 for all v* =I- v E V, the second inequality in (7) follows. One also proves the converse as for Theorem 6.5.12: If J(v, w*) is minimal for v = v* then (5b) holds. (ab) If v* is a solution of (5c), then J(v*, w*) - J(v*, w) = 2[b(w* - w, v*) - h(w* - w)]

vanishes for all w, which proves the first part of (7) in the stronger form J(v*,w) = J(v*,w*).If, however, v* is not a solution of (5c), there exists a wE W such that 6 := J(v*, w*) - J(v*, w) =I- O. Since J(v*, w*) - J(v*, w) = -6 for w := 2w* - w, the first inequality cannot be valid. For the converse define w± = w* 1= w. The first part of (7) implies 0:5 J(v*, w*) - J(v*, w±)

= ±2[b(w, v*) - h(w)]

for both signs, and so b(w,v*) = h(w). Since wE W is arbitrary, one obtains (5c). (ba) We set j(w) := minvEvJ(v,w). According to Theorem 6.5.12, j(w) = J(vw, w), where Vw E V is the solution of (5b). If Vw and Vw' are the solutions for wand WI, it follows that a(vw - vw',x) = F(x) := b(w - wl,x)

Since

IIFllv' :5 Cbllw -

for all x E V.

wlllv and IIvw - vw'llv :5 C/IIFllv' one obtains

IIvw - vw,lIv :5 Cllw - w/llv

for all w, WI E W.

(12.2.9a)

(bb) By using the definition of Vw in (5b) we can write: J(v*, w*) - J(vw, w) = a(v*, v*) + 2b(w*, v*) - 2ft(v*) - 2h(w*) - [a(vw, vw) + 2b(w, vw) - 2ft (vw) - 2h(w)]

= 2[ft(vw - v*) - b(w, Vw - v*) - a(vw, Vw - v*)] + a(vw - v*, Vw - v*) + 2[b(w* - w, v*) - h(w* - w)] = a(vw - v* ,Vw - v*) + 2[b(w* - w, v*) - h(w· - w)]. (12.2.9b) (bc) Let v*, w* be a solution of (5a-c). Because of (5c) the expression in brackets in (9b) vanishes and we have J(v*, w*)

(5b) gives

V w•

= J(vw, w) + a(vw - v*, Vw - v*)

= v*,

~

J(vw, w)

= j(w).

and so J(v*,w*) =j(w*)

= maxj(w)j wEW

(12.2.9c)

282

12 Stokes Equations

i.e., (8) holds. (bd) Now let v*, w* be a solution of (8). If in (9b) one sets w = w*, Vw = Vw' one obtains from J(v*,w*) = j(w*) that v* = Vw" Hence a(vw-v*,vw-v*) = a( Vw - V w' , Vw - vw') depends quadratically on IIw - w* Ilv (cf. (9a». The variation over w := w* - AY (A E 1R, yEW arbitrary) gives 0=

d~j(w* - Ay)/.~=O = 2[b(y, v*) -

f(y»),

and so (5c) is proved. (5b) has already been established with v* =

Vw "

•

12.2.3 Existence and Uniqueness of the Solution of a Saddlepoint Problem To make the saddlepoint problem (5a-c) somewhat more transparent we introduce the operators associated with the bilinear forms: with a(v,x) = (Av,x)v1xv for v,x E V, (12.2.lOa) B E L(W, V'), B* E L(V, W')

A E L(V, V')

= (Bw, x)v'xv = (w, B*X)WXW" (12.2.10b) = (Cu,z)x'xx for u,Z E X. (12.2.lOc) now has the form Cu = f, while (5a-c) can be written

with b(w, x)

C E L(X,X') with c(u,z) Thus problem (6a,b) as

Av+Bw=

11

B*v =/2.

(12.2.11a) (12.2.11b)

If one assumes the existence of A-l E L(V', V), one can solve (lla) for v: (12.2.12a) and substitute in (llb): B* A-1Bw

= B*A- l 11- h·

(12.2.12b)

The invertibility of A is in no way necessary for the solvability of the saddlepoint problem. However, it does simplify the analysis, and does hold true in the case of the Stokes problem. Remark 12.2.5. (a) Under the assumptions

A- l E L(V', V),

(B*A-1B)-1 E L(W', W)

(12.2.13)

the saddlepoint problem (5a-c) [resp. Equations (1Ia, b») are uniquely solvable. (b) A necessary condition for the existence of (B* A -1 B)-l is

B E L(W, V') is injective.

(12.2.14)

12.2 Variational Formulation

283

PROOF. (a) Under the assumption of (B* A-1B)-1 E L(W', W), (12b) is uniquely solvable for w, and then (12a) yields v. (b) The injectivityof B* A-1 B implies (14). • Attention. In general, B: W -+ V' is not bijective so that the representation of (B* A-1 B)-l as B-1 AB*-l is not possible. The example of the 3x 3 matrix C =

(1-1 -1) ( A B). 0 WIth A 1 and BT

=

B =

1 shows that a system (1)

of the fonn (lla,b) can be solvable even with a singular matrix A. Therefore the assumption A -1 E L(V', V) is not necessary. A closer look reveals the subspace V: bey, v) = 0 for all yEW} c V, (12.2.15) which, as we noted before, in general is not trivial. The kernel of a continuous mapping is closed so that V, according to Lemma 6.1.17, can be represented as the sum of orthogonal spaces:

Vo

:= ker B* = {v E V: B*v =

V

O} = {v

E

= Vo ED VJ.. with VJ..:= (Vo)J...

(12.2.16a)

Exercise 12.2.6. Let (16a) hold. Show that (a) V' can be represented as V' = Vti ED Vi, where Vti := {v' E V': v' (v) = 0 for all v E VJ..},

(12.2.16b) (12.2.16c)

Vi := {v' E V':v'(v) = 0 for all v E Vo}.

As a nonn on Vo and Vi one uses II . IIv'. (b) The Riesz isomorphism Jv: V -+ V' maps Vo onto Vo and VJ.. onto Vi. (c) Vo and Vi are orthogonal spaces with respect to 1I·lIv'. (d) The following holds:

IIv'II~, = Ilvtill~'o + Ilvl.lI~'.J..

for v'

= vti +Vl.,vti E Vti,Vl.

E Vi. (12.2.16d)

The decompositions (16a,b) of V and V' define a block decomposition of the operator A: A=

Aoo E L(Vo, Vti),

[1:

AoJ.. E L(VJ.., Vti),

1:~], AJ..o E L(Vo, Vi),

AJ..J.. E L(VJ.., Vi).

Here, for example, Aoo is defined as follows: Aoovo

= vti for

Vo E

Vo if Avo = vti + Vl. with vti E Vti, Vl.

E Vi·

The corresponding decomposition of B* into (Bo, Bl) is written as (0, B*), since = 0 according to the definition of Yo. Conversely, we have range (B) C Vi, so that B = (~). System (lla,b) thus becomes

Bo

284

12 Stokes Equations

Aoovo + Aol. v l. = flO Al.ovo + Al.l. Vl. + Bw = fa B*vl. = h where v

= Vo +Vl., Vo E Yo,

Vl. E Vl., it

= flO + fa,

(12.2.17a) (12.2.17b) (12.2.17c) ito E VcJ, fa E V.f.

Theorem 12.2.7. Let (4a-e) hold. Let Va be defined by (15). A necessary and sufficient condition for the unique solvability of the saddlepoint problem (5a-e) for all f1 E V' is the existence of the inverses AiXl E L(V~, Yo),

B- 1 E L(vI, W).

(12.2.18)

PROOF. (a) (17a-e) represents a staggered system of equations. (18) implies B*-1 = (B-1)* E L(W', Vl.) so that one can solve (17c) for Vl. = B*-1 h. Vo E Va is obtained from (17a): Vo = AO01 UlO - Aol.vl.). Finally, w results from (17b). (b) In order to show that (18) is necessary, we take ito E VcJ arbitrary, fa = 0 and h = O. By hypothesis here we have a solution (vo, Vl., w) E Va X Vl. X W. B*vl. = 0 implies Vl. E Va, so that Vl. = 0 because Vo n Vl. = {o}. Thus Aoovo = flO has a unique solution Vo E Va for each flO E V~. Since Aoo: Va VcJ is bijective and bounded, Theorem 6.1.13 shows that E L(Vo, Va). If one takes fa E Vi arbitrary and flO = 0, h = 0, one infers Vl. = 0 and Vo = 0, so that Bw = fa has a unique solution w E W . .Ai?, we did for A oo , one also infers that B-1 E L(Vi, W). •

AOl

The formulation of conditions (18) in terms of the bilinear forms results in the Babuska-Brezzi conditions: inf{sup{la(vo,xo)l:xo E Va, IIxoliv

= 1}:vo E Va, IIvoliv = I}

~

a > 0, (12.2.19a) sup{la(xo, vo)l: Xo E Va, IIxoliv = I} > 0 for all 0 =I Vo EVa, (12.2.19b) inf{sup{lb(w,x)l:xE V,lIxliv = l}:WE W,lIwllw = 1}~,8> O. (12.2.19c) Exercise 12.2.8. Show that (19a) [resp. (19c») are equivalent to (19a') [resp. (19c'»): for all Vo E Yo, (12.2.I9a') for all wE W. (12.2.I9c')

sup{la(vo,xo)l:xo E Va,lIxoliv = I} ~ allvollv sup{lb(w,x)l:x E V, IIxliv

= I} ~ ,8l1wllw

Lemma 12.2.9. Let (4a,b) hold. Let Vo be defined by (15). Then conditions (18) and (19a-e) are equivalent. Here we have IIAiXlllvo 0

for all 0 =I x e V..L.

(12.2.1ge)

As in the proof of Lemma 6.5.3, we obtain the equivalence of (19a,b) with •

Aol e L(V&, Yo) and of (19d,e) with B-1 e L(Vl., W).

Corollary 12.2.10. (a) Condition (19b) becomes superfluous if a(·, .) is symmetric on Vo x Yo or if Lemma 6.5.17 applies. (b) Each of the following conditions is sufficient for (19a,b) and hence also e L(V&, Yo): for

AoJ

a(·, .): Yo x

Yo ---+ ill. is Yo-elliptic:

a(vo,vo) 2 allvoll~ for all Vo e Yo, a(·, .): V x V ---+ ill. is V -elliptic.

(12.2.20a) (12.2.20b)

PROOF. (a) As in Lemma 6.5.17. (b) (20b) implies (20a); (20a) yields (19a,b). • Exercise 12.2.11. Show that under the assumptions (4a--c) , (18) is also equivalent to the existence of C- l e L(X',X) (cf. (lOc)). Find a bound for IIC-lllx+--x' in terms of IIAolllvo+--v~, IIAllvl+--VI, and IIB-lllw+--vl'

12.2.4 Solvability and Regularity of the Stokes Problem Conditions (19a,b) (Le., problem:

AOol

e L(V&, Yo)) are easy to satisfy for the Stokes

Lemma 12.2.12. Let D be bounded. Then the forms (2a,b) describing the Stokes problem satisfy the conditions (4a,b) and (19a,b). PROOF. (4a,b) is self-evident. According to Example 7.2.10, In('V'u, \7v} dx is HJ(D)-elliptic. From this follows the HA(D)-ellipticity of a(·, .). Corollary lOb proves (19a,b). • It remains to prove condition (19c), which for the Stokes problem assumes the form

SllP{lk w(x)divu(x)dxl:u e HA(D), lull = I} 2 (3lwlo for all w e

L~(D),

(12.2.21)

12 Stokes Equations

286

or

IIVwIIH-l(.a) ~

.8l1wIlL2(0)

for all

w E L~(a).

(12.2.21')

Lemma 12.2.13. Sufficient and necessary for (21) is that for each w E L~(il) there exists 1.1. E H6(il) such that

(12.2.21") PROOF. (a) For w E L~(a) select 1.1. such that (21") holds and set u := u/luh. The left-hand side in (21) is ~ fa w(x) divu(x)dx = Iwl6!luh ~ .8lwlo. (b) If (21) holds one infers as in Section 12.2.3 the bijectivity of B*: V.L ~ W with IIB*-lllv.l.+-w ~ 1/.8. Therefore 1.1.:= B*-lw satisfies condition (21") .

•

Necas [2) proves

Theorem 12.2.14. Condition (21) is satisfied if a E CO,l is bounded. Under this assumption, the Stokes problem

-Llu+Vp=f,

-divu=gina,

1.1.=0 onr

(12.2.22)

has a unique solution (u,p) E H6(n) X L~(il) for f E H-l(a) and 9 E L~(a), with (12.2.23) 11.1.11 + 1P10 $ Cn[ Ifl-1 + Iglo). Remark 12.2.15. Under the conditions that n = 2 and that a E (jl is a bounded domain...the existence proof can be carried out as follows.

PROOF. We need to prove (21"). For w E L~(a) solve -Ll