A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations (Advances in Mechanics and Mathematics)

A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY With Applications in Mo...

Author: Weimin Han

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A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY

A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY With Applications in Modeling and Numerical Approximations

WEIMIN HAN Department of Mathematics University of Iowa Iowa City, IA 52242, U.S.A.

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A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 0-387-23536-1

e-ISBN 0-387-23537-X

Printed on acid-free paper.

O 2005 Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as t o whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1

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Contents

List of Figures List of Tables Preface 1. PRELIMINARIES Introduction Some basic notions from functional analysis Function spaces Weak formulation of boundary value problems Best constants in some Sobolev inequalities Singularities of elliptic problems on planar nonsmooth domains An introduction of elliptic variational inequalities Finite element method, error estimates 2. ELEMENTS OF CONVEX ANALYSIS, DUALITY THEORY 2.1 Convex sets and convex functions 2.2 Hahn-Banach theorem and separation of convex sets 2.3 Continuity and differentiability 2.4 Convex optimization 2.5 Conjugate functionals 2.6 Duality theory 2.7 Applications of duality theory in a posteriori error analysis

3. A POSTERIORI ERROR ANALYSIS FOR IDEALIZATIONS IN LINEAR PROBLEMS 3.1 Coefficient idealization 3.2 Right-hand side idealization

vii xi xv 1 1 5 7 16 20 25 29 36

vi


3.3 3.4 3.5 3.6

Boundary condition idealizations Domain idealizations Error estimates for material idealization of torsion problems Simplifications in some heat conduction problems

4. A POSTERIORI ERROR ANALYSIS FOR LINEARIZATIONS 4.1 Linearization of a nonlinear boundary value problem 4.2 Linearization of a nonlinear elasticity problem 4.3 Linearizations in heat conduction problems 4.4 Nonlinear problems with small parameters 4.5 A quasilinear problem 4.6 Laminar stationary flow of a Bingham fluid 4.7 Linearization in an obstacle problem 5. A POSTERIORIERROR ANALYSIS FOR SOME NUMERICAL PROCEDURES 5.1 A posteriori error analysis for regularization methods 5.2 KaEanov method for nonlinear problems 5.3 KaEanov method for a stationary conservation law 5.4 KaEanov method for a quasi-Newtonian flow problem 5.5 Application in solving an elastoplasticity problem 6. ERROR ANALYSIS FOR VARIATIONAL INEQUALITIES OF THE SECOND KIND 6.1 Model problem and its finite element approximation 6.2 Dual formulation and a posteriori error estimation 6.3 Residual-based error estimates for the model problem 6.4 Gradient recovery-based error estimates for the model problem 6.5 Numerical example on the model problem 6.6 Application to a frictional contact problem REFERENCES

255 262 27 1 287

Index

301

List of Figures

Numerical analysis of a physical problem Smoothness of the boundary Smooth domains Lipschitz domains A crack domain A comer domain near the comer 0 A finite element mesh Continuity of a convex function Subdifferential of the absolute value function Choice of the auxiliary function near a comer domain: Subcase 2A Choice of the auxiliary function near a comer domain: Subcase 2B Domain idealization One-dimensional stress-strain relation Neighborhood around a corner The function for numerical examples Setting of a nonlinear elasticity problem The working problem Example 5.9, convergence of KaEanov iterates Example 5.10, convergence of KaEanov iterates Example 6.11, true solution Example 6.1 1, initial partition and adaptively refined partition after 5 iterations Example 6.11, l/u- u K ~ / / ( 0~ ) ,VS.~ h11211~- Xhllo,r( A )

viii


6.4 6.5 6.6 6.7 6.8

6.9

Example 6.1 1, plots of X h , 1 Example 6.11, plots of Example 6.11, results based on residual type estimator, IIu - ~ x ~ I 1 (01 1 , VS. ~ IIu - utd//1,n (A) Example 6.11, results based on gradient recovery type estimator, llu - uEn (D)VS. llu - utd (A) Example 6.11, results based on residual type estimator, IIu - ~ ; l l ~ l and l ~ ,q~~ on uniform mesh (o) vs. Ilu uidI 11 ,a and q~ on adapted mesh (A) Example 6.11, results based on gradient recovery type estimator, llu - ~ E ~ l and l ~q~, on~ uniform mesh (o) vs. llu and 7~ on adapted mesh (A) Example 6.11, performance comparison between the two error estimators Example 6.12, physical setting Example 6.12, initial mesh with 128 elements, 81 nodes Example6.12, ~ / U - U ; ~( /0/) ~~ sh. 1 / 2 1 1 ~ h T - ~ ~ ~ ~ ~ 0 , r c (A) Example 6.12, plot of X h , ~ Example 6.12, plot of Example 6.12, results based on residual type estimator, deformed configuration 5583 elements, 2921 nodes Example 6.12, results based on gradient recovery type estimator, deformed configuration with 5437 elements, 2832 nodes Example 6.12, results based on residual type estimator, Ilu - u y / / v(0) VS' 1/21 - u;~IIV (A) Example 6.12, results based on gradient recovery type estimator, iiu - u;lln (a) vs. iiu - uid/Iv (A) Example 6.12, results based on residual type estimator, I I U - U ; ~ . ~ I I ~ andqR onuniformmesh(o)vs. IIu-u~d11~ and q~ on adapted mesh ( A ) Example 6.12, results based on gradient recovery type estimator, /lu- u K ~ ( and ( ~ q~ on uniform mesh (o) vs. I 1 u - uid11 and q~ on adapted mesh (A) Example 6.13, physical setting Example 6.13, initial mesh with 160 elements, 105 nodes Example 6.13, results based on residual type estimator, deformed configuration with 5222 elements, 2755 nodes

~ t ~ / / ~ , ~

6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17

6.18 6.19 6.20

6.21

6.22 6.23 6.24

Itv

List of Figures

Example 6.13, results based on gradient recovery type estimator, deformed configuration with 4964 elements, 2624 nodes Example 6.13, plot of Ah,l Example 6.13, plot of Ah,2 Example 6.13, results based on residual type estimator, IIu - ~ ; ~ l(0) l vVS' IIu - u;~IIV (A) Example 6.13, results based on gradient recovery type estimator, Ilu - uKnIIV(o) VS. /Iu - uidlV(A) Example6.13, llu-uKnIIV (o)vs. h 1 / 2 / / ~ h T - ~ K ~ I I 0 , r c (A)

Example 6.13, results based on residual type estimator, I I U - U ; ~ I I ~ andqRonuniformmesh(o)vs. I I U - U ; ~ / ) ~ and q~ on adapted mesh (A) Example 6.13, results based on gradient recovery type estimator, Ilu - uKn11 and 7~ on uniform mesh (o) vs. /lu- uid11 and q~ on adapted mesh (A)

List of Tables

89 Example 3.8, effectivity of error bound 91 Example 3.10, effectivity of error bounds 98 Example 3.12, smallest roots for several angles 140 Example 4.1, true errors and error bounds, X = 0.1 140 Example 4.1, true errors and error bounds, X = 0.015 Example 4.1, true errors and error bounds, X = 0.0 101 140 Example 4.2, error bounds for problems on an L-shape domain 142 Singular exponents Example 4.11, error bounds for various parameters Example 4.12, error bound for energy difference Efficiency of error bounds on a one-dimensional obstacle problem Example 5.7, numerical results with a ( s ) = (2+s)/ (1+ s ) , Po = 718 Example 5.7,numerical results with a ( s ) = (7+s) / ( l + s ) , Po = 114 Example 5.7,numerical results with a ( s ) = (2+s2)/(1+ s2), Po = 7/16 Example 5.8, numerical results with K = 0.20488 Example 5.8, numerical results with K = 10-I Example 5.8, numerical results with K = Example 5.8, numerical results with K = l o w 3 Example 5.8, numerical results with K = Example 5.9, convergence of KaEanov iterates Example 5.10, convergence of KaEanov iterates

xii


5.11 5.12 6.1 6.2

Example 5.11, KaEanov iteration for a quasi-Newtonian flow problem, qo = 1,q, = 0.0001 Example 5-11,KaEanov iteration for a quasi-Newtonian flow problem, qo = 100,q, = 0.01 Nodes and weights of a7-point Gauss-Legendre quadrature formula over the reference triangle Example 6.11, numerical values of constants

226

226 263 268

Dedicated to

Preface This work provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear variational problems. An error estimate is called a posteriori if the computed solution is used in assessing its accuracy. A posteriori error estimation is central to measuring, controlling and minimizing errors in modeling and numerical approximations. In this book, the main mathematical tool for the developments of a posteriori error estimates is the duality theory of convex analysis, documented in the well-known book by Ekeland and Temam ([49]). The duality theory has been found useful in mathematical programming, mechanics, numerical analysis, etc. The book is divided into six chapters. The first chapter reviews some basic notions and results from functional analysis, boundary value problems, elliptic variational inequalities, and finite element approximations. The most relevant part of the duality theory and convex analysis is briefly reviewed in Chapter 2. This brief review is sufficient for the applications of the duality theory in all the following chapters. In mathematical modeling of differential equation problems, usually assumptions are made on various data. Qualitatively, for many problems, it is known that the solution depends continuously on the problem data. Frequently though, it is desirable also to estimate or bound quantitatively the effect on the solutions of the problems caused by the adoption of the assumptions on the data. In Chapter 3, a posteriori error estimates are derived for the effect on the solutions of mathematical idealizations on the data of elliptic linear boundary value problems. In Chapter 4, a posteriori error estimates are given for linearization in a number of nonlinear boundary value problems. The last two chapters are devoted to a posteriori error analysis of numerical solutions. In Chapter 5, the regularization method and the KaEanov method are considered, both being useful in handling certain types of nonlinearity. In Chapter 6, a posteriori error estimates are derived and studied for finite element solutions of some elliptic variational inequalities. This book is intended for researchers and graduate students in Applied and Computational Mathematics, and Engineering. Mathematical prerequisites include calculus, linear algebra, some exposures of differential equations, and concepts of normed spaces, Banach spaces and Hilbert spaces. In the theoretical development, some basic notions and results in functional analysis, duality thoery, weak formulations of boundary value problems, variational inequalities, and the finite element method are used. Brief reviews of these notions and

xvi


results in the first two chapters provide background materials for a reader who lacks knowledge in these areas. This work avoids giving the results in the most general, abstract form so that it is easier for the reader to understand more clearly the essential ideas involved. Many examples are included to show the usefulness of the derived error estimates. In preparing this book, I have benefited from many individuals. I am grateful to Professor Ivo BabuSka for introducing me the research topic and for providing valuable advice. Several of my collaborators (teachers, friends, and students) made contributions to various parts of the book, and I especially thank Dr. Viorel Bostan, Dr. Jiuhua Chen, Professor Hongci Huang, late Professor Sgren Jensen, Professor B.D. Reddy. I express my gratitude to Professor Kendall Atkinson and Professor Mircea Sofonea for their constant support. I thank Professor D.Y. Gao and Professor R.W. Ogden for inviting me to make the contribution in their Kluwer book series on Advances in Mechanics and Mathematics (AMMA). The supports of NSF under grant DMS-0106781 and the James Van Allen Fellowship of the University of Iowa are greatly appreciated.

Chapter 1

PRELIMINARIES

1.1.

INTRODUCTION

Numerical simulation/scientific computation is now playing a more and more important role, and has become one of the three basic tools in science and technology, in addition to experimentation and theory. Numerical simulation provides a relatively inexpensive and efficient way to help understanding the physical world and advancing the technology. A complete numerical analysis simulation session for a physical or engineering problem typically consists of several steps, described below. See Figure 1.1 for a description of the related flow chart, following [ 7 ] . First, the physical or engineering problem is brought to our attention. We want to predict and determine the response of the physical system to the external actions. To do this we need to establish a mathematical model for the problem. This is achieved by applying physical laws, material constitutive relations, and various experimental data such as the geometry of the system, densities of external forces. Most often, we obtain an initial-boundary or boundary value problem of differential equations or differential inequalities to describe the physical or engineering problem. We call this mathematical model the basic mathematical model, and identify it with the physical reality. It is a highly idealized assumption that we can have a mathematical problem which exactly describes the physical problem. The available data, which usually come from experiments, for the basic mathematical formulation can not be obtained as accurate as one wishes. As a consequence, we solve a simplified, or idealized mathematical problem instead. The idealized or simplified problem is the mathematical model we use to study the physical problem. The idealized mathematical model is usually still rather complicated and can be solved only by numerical methods. Popular methods to discretize initial-


Physical Problem 1

Basic Mathematical Model Mathematical Idealization

-f

1

Error ~

n

a

m

4

Output Figure 1.1. Numerical analysis of a physical problem

boundary or boundary value problems include the finite difference method and the finite element method. As a result of the discretization, we obtain a discrete system. The discrete system is then solved by some numerical method. Once we have solved a discrete system, a natural question is: Can we use this solution? In other words, is this discrete solution sufficiently accurate for practical use? The reliability of a numerical solution of a physical or engineering problem depends on mathematical idealization of the physical problem and numerical treatment of the idealized mathematical problem. Various possibilities may arise and they demand our closer investigation. It is desirable to be able to estimate the errors associated with the steps described above. Standard topics in error analysis deal with the errors caused by discretization and solution of the discrete system. Fewer results are available for the estimation of errors in mathematical modelling. The error in mathematical modelling may be the most critical, however. If both the mathematical idealization and the numerical solution of the idealized problem are reliable, various information for the real problem is drawn based on the numerical solution of the idealized problem. If a discrete solution is found to be not accurate enough, we need to trace the sources of the inaccuracy, and decide whether we need to compute a more accurate numerical solution of the mathematical model, or refine and numerically solve a new mathematical model. There is a large amount of literature on numerical methods and their error analysis. Relatively few results are available in the literature for reliability analysis of mathematical idealizations, especially most desirably, certain good, practically useful quantitative assessments of the quality of solutions of idealized problems. Such quantitative assessments should be (hopefully) available once we have computed the solutions of idealized problems. One should not assume the quantitative knowledge of the solutions of basic mathematical problems for either exact descriptions of basic mathematical problems are usually not available in practice or, it is often too expensive to solve the basic mathematical prob-

Preliminaries

3

lems. Some papers devoted to modeling error analysis or error analysis of mathematical idealizations include [30, 71, 72, 116, 118, 119, 120, 121, 128, 1421. An error estimate is called a posteriori if the computed solution is used in assessing its accuracy. A posteriori error estimation is central to measuring, controlling and minimizing errors in modeling and numerical approximations. We now briefly describe the main features of the a posteriori error estimates to be derived and studied in this work. We let (P)stand for the basic mathematical model problem and use u for its solution, and let (Po)be an idealized mathematical problem with the solution uo, and ( ~ ta )numerical approximation of the idealized model with the solution u!. Here, h represents a discretization parameter. The basic mathematical model ( P ) is usually difficult to solve, even numerically, and the simpler problem (Po) is expected to be close to ( P ) . We want to use the solution uo to bound the error l u - uo 11:

where B ( u o ) is a quantity completely computable once uo and some information on the data of ( P ) are known. We also allow the case where the data for ( P ) are not completely given, and only some ranges of the data are available. In such a case, the data for ( P o ) can be obtained through certain averaging process on the data for (P). Of course, to be able to derive a posteriori error estimates, we need to make assumptions on the structure of the problem ( P ) . In this work, ( P ) is assumed to be a convex minimization problem; this allows the employment of the duality theory in convex analysis for deriving a posteriori error estimates. We will also use the numerical solution u! of the problem (P:) to bound the error 11 uo - u! 11. A posteriori estimation of the discretization error (uo - uk) has been a popular research topic since late 1970's (see the description in Chapter 6). Many of the a posteriori error estimates for the numerical solutions of differential equation problems can be derived via the duality theory. In Chapter 6, we focus on the a posteriori error analysis for finite element solutions of elliptic variational inequalities of the second kind. We will see there that the duality approach provides a general framework, leading to various a posteriori error estimators. In this work, errors are measured in terms of energy norms or energy-like norms. The salient features of the a posteriori error estimates presented in this work are: 1 The error estimates are rigorous in the sense that the error bounds are always satisfied. In this sense, when we say error estimates, most often we mean error bounds. We use the phrases "error estimates" and "error bounds" interchangeably.


2 The error bounds are determined by the solutions of the idealized mathematical models or the numerical solutions of the given problems. There is no need to solve related dual problems.

3 All the error bounds, except those presented in Chapter 6 on the finite element approximations of variational inequalities, are completely computable in the sense that there are no unknown constants. In the literature, most a posteriori error estimates on numerical solutions of partial differential equations problems involve such theoretical unknown constants and their values are selected based on a few numerical examples (e.g., 1501).

4 Efficiency of the error estimates is demonstrated through numerous examples and theoretical analysis. Recently, goal-oriented or object-oriented error estimates have been developed to calculate error bounds of global or local quantities of interest, such as the error of stress or strain in a critical region, using a dual-weighted residual technique. The procedure of the technique can be described as follows. Consider a boundary value problem and suppose the purpose of the computation is the value of a functional of the solution. The boundary value problem is solved numerically, typically by the finite element method. The functional of the numerical solution is then computed. To bound the error involved in the functional value, a dual problem is introduced related to the functional and the boundary value problem, and is numerically solved. Then the error in the computed functional value is expressed in terms of the numerical solution of the dual problem together with some residual quantities, which is then localized and split into contributions related to the modeling error and discretization error. The survey papers [19, 621 provide detailed accounts of this technique. Note that the use of solutions of the dual problems may imply a possibly dramatic increase in the computational effort. In this work, dual problems will play a central role in the derivation of error estimators, but the error bounds do not involve solutions of the dual problems. The organization of the book is the following. In the remaining part of this first chapter, we will briefly review some basic notions and results from functional analysis, function spaces, weak formulation of boundary value problems, and the finite element method. In Chapter 2, we review some basic material from convex analysis and the duality theory, that plays the central role in this book for a posteriori error analysis. In Chapter 3, we employ the duality theory to derive a posteriori error estimates for mathematical idealizations in linear boundary value problems, paying particular attention to the situation with nonsmooth domains. The idealizations can occur in the coefficients of the differential equations, the right-hand sides, boundary value conditions, and the domain. In Chapter 4, we perform a posteriori error analysis for the effect of linearization in several nonlinear problems. In Chapter 5, we apply the duality theory to

Preliminaries

5

derive a posteriori error estimates for some numerical procedures in solving nonlinear boundary value problems, including the regularization method for problems involving non-differentiable terms, KaCanov iteration methods and linearizations. Finally, in Chapter 6, we derive a posteriori error estimates for finite element solutions of elliptic variational inequalities of the second kind. The error estimates that can be derived via the duality theory include some of the well-known a posteriori error estimates found in the finite element literature for solving elliptic differential equations.

1.2.

SOME BASIC NOTIONS FROM FUNCTIONAL ANALYSIS

We assume the reader is familiar with such basic notions as linear spaces, norms, inner products, Banach spaces, and Hilbert spaces. Details on these and the material to be reviewed in the following can be found in any standard textbook on functional analysis, e.g., [45, 481, or in a concise form, [6]. In this work, the general theory will be developed for domains in the space IRd of the d-dimensional vectors of the form x = (xl,. . . , x ~ )xi~ E, R. Recall that a domain R c Rd is an open, connected, bounded set in E X d . For p E [I,m], we have the following norms in IRd:

When Rd is viewed as a normed space, implicitly we understand the norm to be the Euclidean norm I . / = / . 12, unless otherwise stated. The Euclidean norm is induced by the canonical inner product in EXd:

The summation convention over a repeated index will be adopted. As an example, for the canonical inner product in IRd, we write (x,y) = xigi. The symbol sdstands for the space of second order symmetric tensors on Rd or, equivalently, the space of symmetric matrices of order d. The inner product and corresponding norm on S d are

For a normed space V ,we use V*to denote its dual space, i.e., the space of all the continuous linear functionals on V .The duality pairing between V* and V is usually denoted by e(v) or (v*,v) for e,v* E V*and v E V .We can then introduce different types of convergence.

6


DEFINITION1.1 Let V be a normed space, V* its dual space. A sequence {u,) C V converges or converges strongly to u E V, written u, + u as n+m,if lim llu - unll = 0. n+cc

The sequence {u,) converges weakly to u E V, written u,

--\

u as n + m, i f

We will use the following property of a weakly convergent sequence:

The dual space of a normed space is always complete, i.e. always a Banach space. Over a finite dimensional space, it is a well-known result that any bounded sequence contains a convergent subsequence. This property does not carry over to infinite dimensional spaces. For example, the sequence {sin j ~ x ) is ~ a>bounded ~ sequence in ~ ~ (I )0, but , none of its subsequences converges. I n many applications of the functional analytic approach, one needs the property that a bounded sequence contains a subsequence that converges in some sense. Reflexive Banach spaces enjoy this kind of desirable property. A space V is said to be rejlexive if (V*)*can be identified with V. A reflexive space must be complete and is hence a Banach space. We have the following important property of a reflexive space.

THEOREM 1.2 If V is a rejlexive Banach space, then any bounded sequence in V has a weakly convergent subsequence. We will see examples of reflexive Banach spaces in Section 1.3. For an inner product, there is an important property called the CauchySchwarz inequality:

with the equality holding iff u and v being linearly dependent. We recall an important property of a Hilbert space.

THEOREM 1 . 3 (Riesz representation theorem) Let V be a Hilbert space, 1 E V*. Then there is a unique u E V for which

In addition, lltll = Il.llv. Thus, the dual space of a Hilbert space can be identified with itself, and any Hilbert space is reflexive.

7

Preliminaries

1.3.

FUNCTION SPACES

We will use the multi-index notation for partial derivatives. An ordered collection of d non-negative integers, a = ( a l , . . , a d ) , is called a multiindex. The quantity ( a (= ai is said to be the length of a. If v is an m-times differentiable function, then for any a with la1 5 m,

c:=,

is the ath order partial derivative. For lower order partial derivatives, there are other notations in common use; e.g., the partial derivative dv/dxi is also written as dx,v, or div, or vjXi,or v,i.

1.3.1

CONTINUOUS FUNCTION SPACES

The notation ~ ( 2 is used ) for the space of functions continuous on 2. It is a Banach space with the norm

More generally, for a non-negative integer m, we define C m ( 2 ) = {v E ~ ( 2 : )D"v E C ( 2 ) for la1

m),

which is a Banach space with the norm

We also set ~ ~ (= 2n;==,crn(2) )

r {v E C ( 2 ) : v

E C m ( 2 ) 'dm = 0,1,. ..)

Given a function v on R, its support is defined to be suppv = {x E R : v ( x ) # 0). Here a bar over a set stands for the closure of the set. We say that v has a compact support if supp v is a proper subset of R: supp v CC R. Thus, if v has a compact support, then there is a strip about the boundary dR such that v is zero on the intersection of the strip and the domain. Later on, we will use the space C,CO(R) = {v E CCO(R): supp v cc 0).

Holder spaces. A function v defined on R is said to be Lipschitz continuous if for some constant c,


8

The smallest possible constant in the above inequality is called the Lipschitz constant of v , and is denoted by Lip(v). More generally, the function v is said to be Holder continuous with exponent /I E ( O , l ] if for some constant c,

(a)

(32) is defined to be the subspace of C which consists The Holder space CO)P of functions Holder continuous with the exponent P. With the norm

the space C O ) P ( ~ ) becomes a Banach space. For a nonnegative integer m and /I E (O,l],we similarly define the Holder space

C~)~= ( E{ V) E crn(a) : D f f vE C

O . ~ ( ~ for )

all u with la. = m } ;

this is a Banach space with the norm

1.3.2

LEBESGUE SPACES

In the study of Lebesgue spaces, we identify functions which are equal a.e. on R. For p E [ I , oo),LP(R) is the linear space of measurable functions v : R -+ R such that

The space L m ( R ) consists of all essentially bounded measurable functions v : R + R such that l l ~ l l l ~ r n= ( ~ )

inf

SUP

Iv(x)I

meas(nl)=O2 ~ n \ n f

< 00.

For a measurable function v defined on R, if v E Lp(Rf)for any R' C C R, then we say v is locally in LP(R) and write v E Lro,(R). We use meas(R)for the Lebesgue measure of R. For d = 3, meas(R) is the volume of R, and for d = 2, meas(R)is the area of R. Some basic properties of the LP spaces are summarized in the following theorem.

9

Preliminaries

THEOREM 1.4 Let R be an open bounded set in IRd. (a) For p E 11, oo],LP(R) is a ~ a n a c hspace with the norm dejined in (1.2)or (1.3). (b) For p E [ I , m ] ,every Cauchy sequence in LP(R) has a subsequence which converges pointwise a. e. on R. (c) If1 5 p 5 q oo, then L Q ( f l )c LP(R),

~

From Definition 1.9, we see that we always have the denseness of the smooth function space Corn ( R ) in w;'P(R). We interpret w;'P(o) to be the space of all the functions v in w k > p ( Rwith ) the "property" that

D a v ( z ) = 0 on d R , V a with ( a (5 k

-

1.

The meaning of this statement will be made clear later after the trace theorems are presented.

Traces. Notice that Sobolev spaces are defined through Lebesgue spaces. Hence Sobolev functions are uniquely defined only a.e, in R. Since the boundary dR has measure zero in Itd,it seems the boundary value of a Sobolev function is not well-defined. Nevertheless it is possible to define the trace of a Sobolev function on the boundary in such a way that for a Sobolev function continuous up to the boundary, its trace coincides with its boundary value.

THEOREM 1.10 Assume R is a Lipschitz domain in IRd, 1 5 p < m. Then there exists a linear operator y : wlJ'(R) -+ LP(dR) such that (a) yv = vIan $v E wlJ'(R) n C ( 2 ) . (b) For some constant c > 0, I l y ~ l l ~ p5( c~~~ w ) i , p b'v ( ~ E) W ' > ~ ( R ) . (c) The mapping y : wlJ'(R) -+ LP(dS2) is compact; i.e., for any bounded sequence {v,) in wlJ'(R), there is a subsequence {v,t) c {v,) such that {yv,t ) is convergent in LP(dR). In Theorem 1.10, the Lebesgue spaces on the boundary, LP(dR),are used. A precise definition of these spaces can be found in [101, Section 6.31. The operator y is called the trace operator, and with property (a) we can view yv as the the generalized boundary value of v. Property (b) states that the mapping y is continuous from w 1 ~ P ( Rto) LP(dR),whereas property (c) further states that actually the mapping y is compact from W1sp(R)to Lp(dS1). The trace operator is neither an injection nor a surjection from w 1 > p ( Rto) LP(dR). The range y (wlJ'(R)) is actually a space smaller than LP(dR),that is denoted by ~ ~ - l l p , p ( d R an) example , of a fractional order Sobolev space. This is a Banach space with the norm

. the future We will frequently use the space H1I2( d ~ =) y ( ~ ' ( 0 ) ) In to simplify the notation, for a function v E w13P((R),we will denote its trace on the boundary also by v. The trace operator is also continuous from

13

Preliminaries

wlJ'(R) to L Q ( R )for q in certain range. One such example is: for d = 2, y E C ( H 1 ( R )H, q ( d R ) )for any q E [I,oo), andforsomeconstantc = c ( R ,q), we have the inequality

With the notion of the trace of wlJ'(R) functions, we have

Here, the condition "v = 0 on dR" is understood as that the trace of v is a zero function on dR. This condition can be equivalently stated as "v = 0 a.e. on do". The particular case p = 2 leads to the Hilbert space

H; ( 0 )= {v E H 1( R ) : v = 0 on 8 0 ) . Its dual space is usually denoted as H-' ( 0 ) . In some occasions, we will need to use the first derivatives (e.g., the normal derivative) on the boundary. Let v = ( v l ,. . . , vd)T denote the outward unit normal to the boundary r of R. Since dR is Lipschitz continuous, v exists a.e. on dR. For a function v E H 2( R ) ,its first derivatives vxi E H 1 ( R ) . By Theorem 1.10, it is meaningful to write their traces vxi E L~( d R ). We can then define the normal derivative

and we have the integration by parts formula

This formula is first proved for u,v E cm(R),and is then extended to u E H 2 ( R )and v E H1( R )by applying the density result Theorem 1.8. For boundary value problems of higher-order partial differential equations, we need to use the traces of partial derivatives on the boundary. Such results can be found in, e.g., [67].

Sobolev embedding theorems. Sobolev embedding theorems are important, especially in analyzing the regularity of a weak solution of a boundary value problem.

DEFINITION 1.1 1 Let V and W be two Banach spaces with V C W . We say the space V is continuously embedded in W and write V v W ifthere exists a constant c > 0 such that

14


We say the space V is compactly embedded in W and write V 3- W , if ( 1 . 5 ) holds and each bounded sequence in V has a convergent subsequence in W. If V 9W , the functions in V are smoother than the rest of the functions in W . Proofs of most parts of the following two theorems can be found in [51]. The first theorem is on embedding of Sobolev spaces, and the second on compact embedding. These theorems are given in a form more general than what is needed later in this work. For any t E R, we denote [t]the largest integer less than or equal to t .

THEOREM 1 . 1 2 Let R c IRd be a Lipschih domain. Then the following statements are valid. (a) I f k < d l p , then wk2P(R) c, L q ( R ) for any q 5 p', where l l p ' = l / p - k/d. (b) I f k = d / p , then w k > p ( Rq ) L q ( R )for any q < cm. (c) I f k > d / p , then

where

4={

[d/pl+ 1- d / p any positive number

p(0) satisfying two conditions:

0 such that

The bilinear form a ( - ,.) is V-elliptic ifthere is a constant m

> 0 such that

and a ( . , .) is symmetric if

a ( u ,v ) = a ( v ,u ) V u , v E V. For a ( . , .) : V x V

+ R a bilinear form and t E V", consider the problem

We have the well-known Lax-Milgram lemma for the existence and uniqueness of a solution of the variational problem (1.17).

THEOREM 1-16 (Lax-Milgram Lemma) Let V be a Hilbert space. Assume a ( . , -) is a bounded, V-elliptic bilinear form on V, t! E V*. Then there is a unique solution to the problem (1.17). Note that the problems (1.9), (1.12) and (1.14) are all of the form (1.17). With the Lax-Milgram lemma, it is easy to show that these problems all admit a unique solution. The V-ellipticity of the associated bilinear form

is deduced from the PoincarC inequality (1.6). The Lax-Milgram lemma can be applied for an existence and uniqueness study of more general linear elliptic partial differential equations. Let R c Rd be a Lipschitz domain, its boundary dR being decomposed as dR = rDU with FD c dfl relatively closed, FN = dR\rD relatively open, and rDnrN= 0.Consider the boundary value problem

19

Preliminaries

Here v = (y, . . . , ud)T is the unit outward normal on rN. We assume the following conditions on the given functions aij, bi, c, f , and 9: aij, bi, c E L m ( 0 ) ; there exists a constant a0

(1.19)

> 0 such that

>

f

a i j t i t j a o 1 ~ 1 ~Y e = (&) E IRd, a.e. in 0; E L2(0), g E L 2 ( b ) .

(1.20) (1.21)

The weak formulation of the problem (1.18) is (1.17) with

We can apply Lax-Milgram Lemma to study the well-posedness of the boundary ~ is a Hilbert space, with the standard value problem. The space V = H ; (0) H 1(0)-norm. The assumptions (1.19)-(1.21) ensure that the bilinear form a ( . , .) of (1.22) is bounded on V, and the linear form is bounded on V. Various sufficient conditions for the V-ellipticity of the bilinear form can be obtained. As an example, denoting b = ( b l , . . . , bd)T, one can verify that any of the following three conditions is sufficient for the V-ellipticity: c or b. v

> co > 0, lbl < B a.e. in 0, and B~ < 4 ~ 0 ~ 0 ,

> 0 a.e. on rN,and c - -21 d i v b > co > 0 a.e, in 0,

or meas(rD) > 0, b = 0, and infc > -ao/c, n where c is the best constant of the PoincarC inequality

A constant E is said to be the best constant of the inequality (1.23) if the inequality holds with c = c and it does not hold for any c < E. The best constant of (1.23) can be computed by solving a linear elliptic eigenvalue problem:

20


with XI

> 0 the smallest eigenvalue of the eigenvalue problem

A special and important case is where b = 0.In this case, the bilinear form is symmetric, and V-ellipticity is assured if

BEST CONSTANTS IN SOME SOBOLEV INEQUALITIES Let R c IRd be a Lipschitz domain, rl a subset of dR with a positive

1.5.

measure. There are a lot of papers and monographs devoted to the study of inequalities among norm quantities such as Ilv 11 L2 Ilv 11 L 2 (rl, 11 V v 11 L 2 ( n ) , 11 dvldvl)L2(r,),//Awl/L2(n)for functions v satisfying various smoothness and auxiliary conditions. For simplicity, in this work, we use IlVv 11 L z c n ) to mean IIVVll(L2(n))d, and we will use both notations. In [I391 one can find a list of 15 inequalities among these quantities for the case when r1= dR. Best possible constants in these inequalities are related to smallest positive eigenvalues of various linear elliptic eigenvalue problems. One example is (1.25) for the best constant (1.24) in the inequality (1.23). The reader is referred to [85, 86, 1031 and references therein for eigenvalue estimations and relations for eigenvalues of different eigenvalue problems. More generally, one can study inequalities between LP-norm of some quantity of v and LQ-normof another quantity of v , for functions v satisfying certain smoothness and auxiliary conditions. When either p or q # 2, the study of the best possible constant in such an inequality no longer leads to a linear eigenvalue problem. For certain specificp and q, we are led to linear elliptic boundary value problems. As an example, we consider the best possible constant in the following trace inequality:

Here R0 is a measurable open subset of R and meas(rl) # 0.This inequality follows directly from the inequality (1.6). We are interested in the computation of the best constant co of the inequality (1.26), characterized by the relation

Let us try to determine this best constant,

Preliminaries

First we have an existence result for the best constant.

L E M M A1.17 There exists 0 f u E H : ~( R )such that

Proof. We choose a sequence {u,) C H i l ( R )with the properties

and I / u n I I ~ l ( ~-o ))CO

as n

--?,

m.

Since {u,) is bounded in H& (O), we can apply Theorem 1.2 on the reflexive Banach space H : ~( R )and Theorem 1.13 for the compact embedding of H& ( 0 ) in L ' ( R ) to conclude the existence of a subsequence {un,} and u E H+l ( 0 ) such that

Thus we have (cf. (1.1))

Therefore,

.

By the definition of the constant co, the above inequality is an equality. Now we perform a formal calculation to derive a formula for the computation of co and then prove the formula rigorously. Let u E Hhl ( 0 )be the function specified in Lemma 1.17. For any v E Hhl ( R )the real function

is defined in a neighborhood of t = 0 and has its maximum at t = 0. We fixed, then the value 11 Qu I I L 2 ( n ) observe that if we keep the value of 11 u 11 L~


will be minimal when the function u does not change rapidly. So we expect an optimal function u should keep its sign. If this is true, then we will have

Hence, if we normalize u with the condition

then u E H& ( f l ) is the solution of the problem

This problem has a unique solution by the Lax-Milgram lemma (Theorem 1.16). So the best constant is expected to be

Let us show that these formulas indeed provide the best constant. For this purpose we need a preliminary result (see, e.g., [61]).

LEMMA1.18 I f v E H 1 ( f l ) ,then Ivl E ~ ' ( f l and ),

i

Vv

Vlvl =

0 -Vv

i f v > 0, i f v = 0, i f v < 0.

From this lemma, we immediately obtain the next result.

LEMMA1.19 I f v E H;l ( f l ) ,then J v JE Hhl ( f l ) and

THEOREM 1.20 Let u E H i 1( 0 )be the solution of the problem (1.27). Then the best constant co of the inequality (1.26) is given by

Proof. For any v

E

H;; (O), by Lemma 1-19,we have Iv 1 E Hhl ( f l ) and

Preliminaries

Therefore,

This completes the proof. Later in Subsection 3.5.2, we will use Theorem 1.20 in the derivation of an a posteriori error estimate in the study of material idealization for a torsion problem in linearized elasticity. We now examine some concrete examples.

EXAMPLE 1 . 2 1 Let d

> 2. Assume

is the d-dimensional open ball of radius Ro centered at 0, Ro = B!:) is the ro Ro, and rl= The d-dimensional ball of radius ro centered at 0, related problem (1.27) is the weak form of the boundary value problem:

2, the solution is "I2

--

u(x) =

+

2d rf d (d - 2)

3 2d

+

o'd

d ( d - 2)

(ri-d-~i-d) i f 0 5 1x1 < T O ,

~i-~)

( 1 ~ l 2 - ~

i f r o 5 1x15 RO


24 and so

Here r ( . ) denotes the gamma function (cf. e.g., [104]). Hence, we have the optimal inequality

and

In particular, when ro = Ro, we have the optimal inequality

for any dimension d 2 2.

rn

EXAMPLE 1.22 Assume R = Ro = n $ , ( 0 , a j ) , T1 = dR. The corresponding problem (1.27) is the weak form of the boundary value problem

The solution u ( x )of the boundary value problem is

Hence, the best constant co = I ~ V U I Iof~the ~~ corresponding ~) Sobolev inequality equals

25

Preliminaries

For a general domain, it is usually impossible to find a closed form for the best constant co. In this case, numerical methods can be used to determine the constant. An example of this kind can be found in [68]. Depending on the applications, best constants of other Sobolev inequalities may be useful. In [70], the following trace inequality is considered:

where Fl and r2 are disjoint subsets of d R with positive surface measures. Then the optimal constant is shown to be

where u E

HI!,

( 0 )satisfies the weak formulation

Discussion of the best constants in some more Sobolev inequalities can be found in [73].

1.6.

SINGULARITIES OF ELLIPTIC PROBLEMS ON PLANAR NONSMOOTH DOMAINS

The most significant property of elliptic boundary value problems on a smooth domain is the so-called "shift theorem". Roughly speaking, "shift theorem" states: For an elliptic boundary value problem on a smooth domain, the smoother the data (the coefficients, the right-hand side and the boundary value), the smoother the solution. As an example, consider the boundary value problem

Assume d R is smooth. Then for any integer k 2 0, f E H k ( R ) implies uE H~+~(R). On nonsmooth domains, however, such a nice property no longer exists. To explain this, consider the two dimensional model problem

We assume that d R is smooth everywhere except at the origin 0, in a neighborhood of which, R coincides with a cone


Figure 1.6. A comer domain near the comer 0

where w is the internal angle of R at 0; see Figure 1.6. Let a = ~ / w For . each positive integer k , define

r k f fsin k a 8 r k f f( l n r sin k a 8

+

if k a # integer, 8 cos k a 8 ) if k a = integer.

+

Then i f f E WmlP(f2) and m 2 - 2 / p is not an integer, we have the following smoothness property for the solution u (cf, citeGr):

for some constants c k , which are certain linear functionals of f . Hence, no matter how smooth the function f is, the smoothness of the solution u is determined by the smoothness of the singular term ul as long as cl = cl ( f ) # 0. We note that ul E w11p(R)if and only if 1 < a 2 / p . An early systematic study of singular behavior around a comer for the solution of an elliptic problem was done by Lehman, cf. [105]. The model problem considered in that paper is the Dirichlet problem for the equation

+

where R is an open planar domain and part of its boundary d R consists of two analytic arcs rl and r2,which meet at the origin and form there an internal

27

Preliminaries

angle w > 0. Assume ri(i = 1 , 2 ) is an analytic curve of which the origin 0 is a regular point. Let u E C 2 ( R )n C ( R )be a solution of the equation with boundary conditions

Assume A, B, C , f , cpl and cp2 are analytic in a neighborhood of the origin and w ( 0 )= cp2 (0) - Then

where the regular part U R together with its partial derivatives of all orders remain bounded when x + 0,and with a = T/W,

u s ( x )=

a) {W O(P r) ln

if a # integer, if a = integer.

These relations may be formally indefinitely differentiated. In particular, if f = 0, cpl = 0, cp2 = 0 in a neighborhood of 0 ,then for some constant k ,

u ( x ) = k r" sin a0

+ o(ra)

and this relation can be formally differentiated. Hence,

Later, Wigley [I561 extended Lehman's results to allow nonanalytic data and boundary conditions involving first order derivatives. In [96], Kondrat'ev developed a general method for studying singular behaviors of solutions of elliptic boundary value problems in the neighborhood of a singular point of the boundary. The influence of the paper on later researches on the topic is tremendous. Maz'ya and his co-researchers extended Kondrat'ev's results in several aspects, e.g., from L~ estimates to Lp estimates for any p E (1,m), from single equations to systems, etc. The interested reader is referred to [107]. More recent comprehensive references on the topic are [99, 1001. In the literature, one may find many other results on the topic. In [146], [157], etc., theoretical and computational aspects of singularities for elasticity systems are studied. The papers [92] and [I231 are devoted to a study of regularity of solutions of Stokes problems in a polygon. For boundary value problems of the biharmonic equation on comer domains, see [23], etc. For both edge and corner singularities in a three dimensional polyhedron, cf. [126], etc. There are also several papers devoted to singular behaviors of solutions of quasilinear elliptic problems on nonsmooth domains, cf. [log], [149], etc. The reader is referred to the bibliographies of [46], [67], 1971, [99] and [loo] for a relatively complete list of references on singularities caused by the nonsmoothness of domains.

28


For singular behaviors of solutions of elliptic problems on nonsmooth domains, the main reference used here is [67]. This reference also provides detailed discussions of Sobolev spaces over domains with piecewise smooth boundaries. The following result is Theorem 5.1.3.5 in [67]. Let R be a polygon, the boundary of which is the union of a finite number 1 5 j N, listed in a counterclockwise order. Split of linear segments the set { 1 , 2 , . . . , N) into two subsets N and V.Let u E H I ( 0 )be a weak solution of the boundary value problem

c,

0 on re.We observe that (1.33) is the constitutive relation of the linearized elasticity material, (1.34) defines the linearized strain and the displacement, (1.35) is the equilibrium equation. With a = (aij)dXd, D i v a : R + Ktd is defined by


34

where the summation convention is used. The classical displacement and traction boundary conditions are given in (1.36) and (1.37). Contact conditions are described in (1.38)-(1.41). The bilateral contact feature is reflected by (1.38). The relations (1-39)-(1.41) represent Tresca's friction law. In certain situations, the frictional contact problem stated above describes the material deformation quite accurately. In more complicated situations, such as when the contact zone is not prescribed a priori or when more realistic frictional contact laws are used, the frictional contact problem here can be viewed as an intermediate problem for a typical step in an iterative solution procedure for solving the more complicated contact problem. For a variational analysis of the problem, we need to introduce a function space and some functionals defined over the space. Let

Its inner product and norm are chosen to be

Since meas(FD) > 0, the Korn inequality (cf. e.g. [47] for its proof)

lv

is a norm over V which is equivalent to the canonical holds. Hence, llv norm 11v 111,fiOver the space V, we define

To derive the corresponding variational inequality, we temporarily assume the problem (1.33)-(1.41) has a solution u which is sufficiently smooth so that all the following calculations are valid. We multiply the differential equation (1.35) by (v - u)with an arbitrary v E V,

35

Preliminaries

We then perform an integration by parts for the left hand side, and use the boundary conditions (1.36) and (1.37) and the constitutive relation (1.33),

+ jflC E( u ). E ( V- u ) dx. Now consider the integral on r c . From the contact conditions (1.39)-(1.41), we have u , . u , = -gIuTl o n r c . This can be deduced by distinguishing two cases: la, / Hence, using the decomposition

0, the bilinear continuous form a ( . , .) : V x V -+ R is V-elliptic, following the Korn inequality (1.42). It can be easily checked that t : V -+ R is a linear continuous functional and j : V -+ is a proper lower semicontinuous convex functional. Therefore, by Theorem 1.25, the variational inequality (1.46) has a unique solution u in V . Moreover, since the bilinear form a ( . , .) is symmetric and positive definite,

36


solving the variational inequality (1.46) is equivalent to minimizing the energy functional 1 E(v) = -a(v,v) - t(v) j(v) 2 over the space V. The equivalence between the variational inequality (1.46) and the minimization formulation can be shown by an argument similar to that used in Example 1.26. Note that the convexity of j (.) implies

+

f o r a n y u , ~E V andandt E [ O , l ] . Variational inequality formulations of many other contact problems can be found in [94, 811.

1.8.

FINITE ELEMENT METHOD, ERROR ESTIMATES

Weak formulations of boundary value problems are the basis for development of Galerkin methods, a general framework for approximation of variational problems, that include the finite element method as a special case. Consider the problem (1.17) (page 18) again. In general, it is impossible to find the exact solution of the problem (1.17) because the space V is infinite dimensional. A natural approach to construct an approximate solution is to solve a finite dimensional analog of the problem (1.17). Let VN c V be an N-dimensional subspace. We project the problem (1.17) onto VN,

Under the assumptions that the bilinear form a ( - ,.) is bounded and V-elliptic, and t E V', we can again apply the Lax-Milgram Lemma and conclude that the finite dimensional approximation problem (1.47) has a unique solution u ~ . The approximate solution U N is, in general, different from the exact solution u.To increase the accuracy, it is natural to seek the approximate solution u~ in a larger subspace VN. Thus, for a sequence of subspaces VN, c VN2 c - - - c V, we compute a corresponding sequence of approximate solutions u r ~E ~VNi from (1.47) for i = 1 , 2 , . . This solution procedure is called the Galerkin method. In the special case when the bilinear form a ( . , .) is also symmetric, a

a ( u , v) = a(v, u)

Vu, v E V,

the original problem (1.17) is equivalent to a minimization problem uEV,

E ( u ) = inf E ( v ) , vEV

(1.48)

Preliminaries

where the "energy functional" is defined as

Now with a finite dimensional subspace VN c V chosen, it is equally natural to develop a numerical method by minimizing the energy functional over the finite dimensional space VN, UN

E V N , E ( u N ) = inf E ( v ) . VEVN

Following the argument for the equivalence of (1.31) and (1-32)in Example 1.26, we can verify that the two approximate problems (1.47) and (1.50) are equivalent. The method based on minimizing the energy functional over finite dimensional subspaces is called the Ritz method. From the above discussion, we see that the Galerkin method is more general than the Ritz method, while when both methods are applicable, they are equivalent. Because of this, the Galerkin method is also called the Ritz-Galerkin method. Once a numerical method is formulated, an important issue is convergence and error estimation for the method. In this regard, for the Galerkin method, a key result is the following CCa's inequality.

PROPOSITION 1.28 Assume V is a Hilbert space, VN C V is a subspace, a ( . , is a bounded, V-elliptic bilinear form on V, and t? E V*. Let u E V be the solution of the problem (1.17),and U N E VN be the Galerkin approximation deJined in (1.47). Then there is a constant c such that a )

The proof of Cia's inequality follows from the error relation

obtained by subtraction of (1.47)from (1.17)with v E VN, and the boundedness and V-ellipticity of the bilinear form a(., .). CCa's inequality is a basis for convergence analysis and error estimations. As a simple consequence, we have the next convergence result.

COROLLARY 1.29 We make the assumptions stated in Proposition 1.28. Assume VN, c VN, c - . . is a sequence of subspaces of V with the property

Then the Galerkin method for (1.17) converges.

38


in V. Cia's In Corollary 1.29, Ui>lVN, stands for the closure of Ui>lVN, inequality also serves a s a basis for error estimates. When the finite dimensional space VN is constructed from piecewise (images of) polynomials, the Galerkin method leads to a finite element method. In other words, the finite element method (FEM) is a Galerkin method with the use of piecewise (images of) polynomials over a finite element partition of the domain 0. The finite element method today is the dominant numerical method for solving problems in structural mechanics and is popularly used in fluid mechanics. It is widely applied to both linear and nonlinear problems. General mathematical theory of finite element methods can be found in [8,41,43,117,144], among others. The textbook [89] offers an easily accessible mathematical introduction of finite element methods, whereas the two recent textbooks, [31, 321, provide deeper mathematical theory together with more recent and current research development such as the multigrid methods. Traditionally, convergence of finite element solutions is achieved through mesh refinement with the use of piecewise low degree polynomial. Since h is usually used to denote the mesh size, the traditional finite element method is also termed as the h-version finite element method. Convergence of the method can also be achieved by using piecewise increasingly higher degree polynomials over relatively coarse finite element meshes, leading to the p-version finite element method. Detailed discussion of the p-version finite element method can be found in [146]. The p-version approximation is more efficient in areas where the solution is smooth, so it is natural to combine the ideas of the p-version and the h-version to make the finite element method very efficient on many problems. A well-known result regarding the h-p-version finite element method is the exponential convergence rate for solving elliptic boundary value problems with corner singularities. Comprehensive mathematical theory of the p-version and h-p-version finite element methods with applications in solid and fluid mechanics can be found in [137]. Mixed and hybrid finite element methods are often used in solving boundary value problems with constraints and higher order differential equations. Mathematical theory of these methods can be found in [33,135]. Several monographs are available on the numerical solution of Navier-Stokes equations by the finite element method, see e.g. [63]. Theory of the finite element method for solving parabolic problems can be found in [I471 and more recently in [148]. Finally, we list a few representative engineering books on the finite element method, [20, 88, 163, 1641. The reader is referred to two historical notes [115, 1621 on the development of the finite element method. In this section, we will review some results of the finite element method. There are some basic aspects in the construction of finite element approximations. First we need a partition (or triangulation) of the domain of the differential equation into sub-domains called elements. Associated with the partition, we

Preliminaries

Figure 1.7. A finite element mesh

define a finite element space. Finally, we choose basis functions for the finite element space. The basis functions should have small supports. For simplicity, we assume R is a two-dimensional polygonal domain. The reader is referred to the aforementioned references for some detailed discussion on the use of the finite element method where R is not polygonal, or where the spatial dimension is higher than two.

Triangulation. A triangulation, partition, or mesh is a partition Ph = { K ) of the polygonal domain 2 into a finite number of subsets K , called elements, with the following properties: -

R = U K E P hK . 0

Each K is a triangle or quadrilateral with a nonempty interior K . For distinct K 1 ,K z E Ph, K 1 f l K 2 is either empty, or a common vertex, a common side or a common face of K l and K 2 . For any K E Ph, its diameter diam ( K ) 5 h. The assumption that the domain R is a polygon ensures that it can be partitioned into straight-sided triangles and quadrilaterals. See Figure 1.7 for a finite element mesh. For convenience in practical implementation as well as in theoretical analysis, it is assumed that there exist a finite number of fixed polyhedra, called reference elements or master elements, ambiguously represented by one symbol K ,such that for each element K , there is an invertible affine mapping function FK with K = FK ( K ). In the two-dimensional case, it is customary to use an equilateral or right isosceles triangle for triangular elements, and squares with side length 1 or 2 for quadrilateral elements.

40


For an arbitrary element K , we introduce its diameter

and let p~ be the diameter of the largest sphere inscribed in K. When dealing with the reference element K we denote the corresponding quantities by h and @. The quantity h K describes the size of K , whereas the ratio h K / p K is an indication whether the element is flat: the larger the ratio h K / p K ,the flatter the element K. For a triangulation P h , we denote h = max h K , KEP,

often called the mesh parameter. The quantity h is a measure of how refined the mesh is. The smaller h is, the finer the mesh. We say a family of triangulations { P h ) h is regular if 1 there exists a constant a 2 1 such that

2 the discretization parameter h

+ 0.

A condition equivalent to the first requirement is the minimal angle condition, i.e, the minimal angles of all the elements are bounded below away from 0.

Polynomial spaces on the reference elements. Function spaces over a general element can be constructed from those on the reference element. Thus we first introduce a polynomial space x on K. Although it is possible to choose other finite dimensional function spaces as X, the overwhelming choice of x for practical use is a polynomial space, due to the simplicity in constructing a polynomial and the good approximation property of polynomial spaces. Since the highest spatial derivatives are of order 2 for the mathematical problems studied later in this work, it is natural to use Lagrange finite elements, e.g., the parameters of the elements are function values at the nodes. Generally, let x be a polynomial space over K, dim^ = No, and we choose a set of nodal points { 8 i ) 2 1 , called the nodes, in K such that any function d E x is uniquely determined by its values at . Then we have the formula

{iii)2,

Nn

where the functions

{&)Fl form a basis for the space x with the property

41

Preliminaries

Affine-equivalent finite elements. We then define a function space over a general element K that is the image of the reference element K under an invertible affine mapping

The mapping FK is a bijection between K and K. Over the element K ,we define a finite dimensional function space X K by

Since FK is an invertible affine mapping, if x is a polynomial space of certain degree, then X K is a polynomial space of the same degree. Any function v defined on K is associated with a function 6 defined on K through .ir = v o FK. We see that v = 6 o Fil. Thus we have the relation

~ ( x=) 6 ( k ) V x E K ,2 E K ,w i t h x = FK(ii) Using the nodal points h i , 1 5 i 5 No, of K ,we can define the nodal points on K : aiK = F K ( h i ) , i = I , . . . , N O . (1.54) Recall that { & ) z , are the basis functions of the space x associated with the nodal points { h i ) F l with the property that

We define

#f=&oFil,

i=1,. . . , No.

Then the functions { @ ) Z 1 have the property that

Hence,

{@)zl

form a set of basis functions for the space X K

Finite element spaces. A global finite element function vh is defined piecewise by the formula v h l ~E X K V K E Ph. A natural question is whether such finite element functions can be directly used to solve a boundary value problem. For a linear second-order elliptic boundary value problem, the space V for the corresponding weak formulation is a subspace of H 1 ( R ) . Since the restriction of vh on each element K is a smooth function, a necessary and sufficient condition for vh E H 1 ( R ) is

42


vh E C ( 2 ) (cf. [41]). Thus we need to check if vh E ~ ( 2 holds. ) We then define a finite element space corresponding to the triangulation Ph, We observe that if x consists of polynomials, then a function from the space X h is a piecewise image of polynomials. In our special case of an affine family of finite elements, FK is an affine mapping, and vh 1 is a polynomial. We can use the finite element space Xh to approximate the space H 1( R ) . Some boundary value problems involve essential boundary conditions. Then we need finite element sets to approximate subsets of H1( R ) . As an example, suppose rl c r is a relatively closed subset of the boundary ?? consisting of a finite number of line segments. Consider the space We construct finite element partitions of R that are compatible to the subset r l , i.e., if K is a finite element, then either K n rl = 8 or K f l r c rl. Then we define the finite element space to be Vh = V n Xh. In other words,

For an affine family of finite elements, the mesh is completely determined by the reference element K and the family of affine mappings {FK : K E Ph). The finite element space Xh is completely described by the space x on K and the family of affine mappings.

Interpolation. On any element K , we introduce an interpolation operator l l K by

i=l We see that n K v E X K is uniquely determined by the interpolation conditions

For a function v E ~ ( 2 1 we,construct its global interpolant n h v in the finite element space X h by the formula

Let { a i ) z l c 2 be the set of the nodes collected from the nodes of all the elements K E Ph. We have the representation formula

43

Preliminaries

for the global finite element interpolant. Here $i, i = 1 , . . . , Nh, are the global basis functions that span X h . The basis function 41i is associated with the node ai, i.e., $i is a piecewise polynomial, $i l K E X K , and $i ( a j )= S i j . If the node ai is a vertex a r of the element K , then $i 1 = $ r . If ai is not a node of K , then $ilK = 0. Thus the functions $i are constructed from local basis functions .4:

EXAMPLE 1.30 We examine an example of linear elements. Assume R c IR2 is a poly onal domain, which is triangulated into triangles K , K E P h . Denote by {ai}$;' the set of the interior nodes, i.e. the vertices of the triangulation that the set of the boundary nodes, i.e. the vertices of the lie in 12; and {ai}2Nint+l triangulation that lie on 8 0 . From each vertex ai, construct K~ as the patch of the elements K which contain ai as a vertex. The basis function q!~~ associated with the node ai is a continuous function in 2,which is linear on each K and . corresponding piecewise linear function space is is non-zero only in K ~ The then X h = span{41i, 1 5 i 5 Nh}. Suppose we need to solve a linear elliptic boundary value problem with Neumann boundary condition. Then the function space is V = H ' ( R ) , and we choose the linear element space to be Vh = Xh. Now suppose the homogeneous Dirichlet boundary condition is specified. Then the function space is V = H: ( R ) ,whereas the linear element space is

In other words, we only use those basis functions associated with the interior nodes, so that again the finite element space is a subspace of the space for the boundary value problem. In case of a mixed boundary value problem, where we have a homogeneous Dirichlet boundary condition on a relatively closed part of the boundary rl and a Neumann boundary condition on the remaining part of the boundary r 2 , then the triangulation should be compatible with the splitting 8R = rl U r 2 , i.e., if an element K has one side on the boundary, then that side must belong entirely to either rl or E. The corresponding linear element space is then constructed from the basis functions associated with the interior nodes together with those boundary nodes located on PI. rn

Finite element error estimates. Let PI,( K )denote the space of the polynomials of degree less than or equal to k on K.A basic result concerning finite element interpolation errors is the following.

+

T H E O R E1.3 M 1 Let k and m be nonnegative integers with k > 0, k 1 2 m, and pk ( K ) C X . Let IIK be the operators defined in (1.55). Furthermore,

44


assume { P h ) is a regular family ofjnite element partitions. Then there is a constant c such that for m 5 k 1,

+

Since the global finite element interpolant is defined piecewise in terms of local finite element interpolants, we have the next result for global finite element interpolation error estimates.

T H E O R E1.32 M Assume that all the conditions of Theorem 1.31 hold. Then there exists a constant c independent of h such that for m = 0,1,

Notice that for the Lagrange finite elements, IIhv @ H 2 ( R ) .For 2 5 m 5

k

+ 1, (1.59) is replaced by

Cia's inequality and the interpolation error estimates together lead to convergence order error estimates for finite element solutions of the elliptic boundary value problem (1.17). As a sample result, consider the finite element approximation of the problem (1.9) (page 16). Assume R c Kt2 is a polygonal domain, f E L2( R ) . Let { P h ) be a regular family of triangulations of into triangles and let { V h ) be the corresponding family of linear element spaces given by (1.57). Under the ) if R is convex, cf. Section solution regularity assumption u E H ~ ( R(valid 1.6),then for the corresponding finite element solutions { u h ) ,we have the error estimate (1.60) 1 1 -~ ~ h l I 1 5, ~c 1 1 -~ n h ~ l l l , R5 c h iul2,R. This estimate is an example of an a priori error estimate , i.e., we have this estimate (under the stated assumptions) before we actually compute the solution uh. The estimate (1.60) expresses the fact that the convergence order of the linear element solutions is one, and if we refine the triangulation by connecting the three side mid-points of each element and compute the new finite element solution uh/2, then roughly speaking, we can expect the error ilu - uh/2 11 1 , would be about half of the error Ilu - uhiIlls2,at least when h is sufficiently small. Note, however, that the a priori error estimate (1.60) does not provide quantitative information on the size of the error 11 u - uh /I l , ~nor , does it provide information on the distribution of the error ( u - u h )throughout R. In contrast, in an a posteriori error estimate, computed solutions are used to provide such quantitative information on the solution errors.

~

Preliminaries

45

In a posteriori error analysis of finite element solutions (cf. Chapter 6), one needs finite element interpolations of functions with low degree smoothness, such as H 1(R) functions. Note that for d 2 2, an H' (R) function can have discontinuity and so the ordinary finite element interpolation based on the use of pointwise function values is not defined. In Section 6.1, we review the definition and some error estimates for one generalized finite element interpolation operator defined for H' (0)functions.

Chapter 2

ELEMENTS OF CONVEX ANALYSIS, DUALITY THEORY

The a posteriori error estimates presented in this work are derived based on the duality theory of convex analysis. The first research monograph specifically devoted to the topic of convex analysis is [136], emphasizing the finitedimensional case. Convex analysis and duality theory in general normed spaces, mostly infinite dimensional ones, are thoroughly discussed in the well-known reference [49]. Another comprehensive treatment of the topic is [159]. Duality theory has been also extended for nonconvex systems, see, e.g. [59, 601 where the mathematical theory is motived by duality in natural phenomena with particular emphasis on mechanics. In this chapter, we review some basic notions and results on convex sets, convex functions and their properties as well as the duality theory. Detailed discussions and proofs of the stated results can be found in [49] or [159]. In the theory of convex analysis, it is convenient to consider functions that take on values on the extended real line E. Recall that a functional f : V + is said to be proper if f ( v ) > -cc b'v E V and f ( u ) < cx for some u E V.

2.1.

CONVEX SETS AND CONVEX FUNCTIONS

Let V be a linear space.

DEFINITION 2 . 1 A subset K C V is said to be convex if it has the property U,V

( I - t ) u + t v E Kb't E [O,l].

EK

We see that if K is convex and u, v E K , then the line segment connecting u and v, i.e, the set ( ( 1 - t ) u t v : t E [0,I ] ) ,is contained in K. By an induction argument, for any ul , . . . ,u, E K and any nonnegative numbers tl , . . . , t , with C:='=,ti = 1, we have C:=l tiui E K . The expression C:=ltiui with

+

48


nonnegative numbers t l , . . . , tn satisfying Cy=2=1 ti = 1 is called a convex combination of the elements u l , . . . , u,.

DEFINITION 2 . 2 Afunction f : V

-+

is convex if

for any u, v E V and t E (0,I ) for which the right hand side is meaningful, i.e., f ( u )and f ( v )are not simultaneously infinite with opposite signs.

DEFINITION 2.3 Let K be a convex set in V avtd f : K

f ( v )=

+R. If

{ y; w

v E K, v

is convex, then we say f is convex on K. Thefunction f is strictly convex on K ifthe strict inequality in (2.1) holds for any u, v E K , u # v and t E ( 0 , l ) . In the future, for a function f defined on a subset K c V, we identify it with its extension f introduced in Definition 2.3. In other words, we will use the same symbol f for both the function defined on K and its extension by oo to the complement of K in the space V. Thus, we will say that a function is convex over a subset K c V to mean that the extension of the function is convex in the space V. By an induction argument, i f f is convex over a convex set K , then we have

for any u l , . . . , u, E K and any nonnegative numbers t l , . . . , t , with

The next result follows easily from the definition of a convex function.

PROPOSITION 2.4 Let V be a linear space, A be an index set. Assume g, t f ( t E f , g , fa ( a E A) : V + are convex. Then the functions f (0,oo)),sup{f , g ) and supaEAf a are all convex. Here we let f ( v )+ g ( v )= oo i f f ( v )= -g(v) = &oo.

+

In the study of convex functions, it is convenient to use the notions of the effective domain and the epigraph.

DEFINITION 2 . 5 Given afunction f : V

-+ E, we dejne its effective domain

dom ( f ) = {v E V : f ( v ) < oo}

Elements of convex analysis, duality theory

and its epigraph e p i ( f ) = { ( v , a )E V x R :f ( v ) 5 a ) . It is easy to show that for a convex function the effective domain is a convex set in V and the epigraph is a convex set in V x R. From now on, we assume V is a normed space.

DEFINITION 2.6 A function f : V -+ is said to be lower semicontinuous (1.s.c.) iffor any sequence {u,) C V with u, -+ u in V , f(

u ) < lim inf f (u,). - n+m

There is a useful characterization of the lower semicontinuity that provides an alternative definition of 1.s.c. in some references.

PROPOSITION 2.7 Thefinction f : V { v E V : f ( v ) 5 r ) is closed.

-+@ is 1.s.c. ifffor any r E R the set

Later on, we will also need the notion of weak 1.s.c.

-

DEFINITION 2.8 A function f : V -+ is said to be weakly lower semicontinuous (w.1.s. c.) i f for any sequence {u,) c V with u, u in V , f(

U)

< lim inf f - n-+m

(u,).

EXAMPLE2 . 9 Let K C V. The indicator function of the set K is defined by

Then it can be verified that K is closed iff IK is l.s.c., whereas K is a convex set iff IK is a convex function. 0

For a set K in the normed space V, we use int K = i n t ( K ) = K to denote its interior, i.e. the set of the points in K such that each point is contained in an open ball that in turn lies in K. Roughly speaking, int K is the subset of K excluding the boundary points. Some of the boundary points may not belong to K , unless K is a closed set. We use K to denote the closure of K , i.e., the union of the set K and its boundary. The following results are not difficult to prove.

PROPOSITION 2.10 Let f : V + E. Then (a) f is convex z f f epi (f ) is convex; (b) f is 1,s.c, ifSepi ( f ) is closed; (c) f is continuous at u and f ( u )# fcx ==+ int epi ( f ) # ( 4 f $ +W ===+ epi ( f ) # 0: (e) f is convex =+ dom (f) is convex.

0;

50

2.2.


HAHN-BANACH THEOREM AND SEPARATION OF CONVEX SETS

The Hahn-Banach theorem and its corollaries are of central importance in functional analysis (cf. e.g. [48]). In this work, we only need Corollary 2.17 given at the end of the section. For completeness, we state some related results and show how they lead to a proof of Corollary 2.17.

DEFINITION 2 . 1 1 A function p : V -+ R is sublinear i f

We observe that p : V -+ R is a seminorm if it is sublinear and p ( t v ) = It/p(v) for any v E V and any t E R.The analytic form of a general HahnBanach Theorem is the following.

THEOREM 2.12 (Hahn-Banach Theorem) Let V be a real linear space, K C V a subspace. Assume f : K -+ R is linear and f (v) 5 p(v) for any v E K, with some sublinearfunctional p : V -+ R. Then f can be extended to a linear functional f : V -+ E% such that f (v) I p(v) for any v E V . Taking the functionp(.) to be a constant multiple of the norm, we immediately get the usual form of the Hahn-Banach Theorem.

COROLLARY2 . 1 3 Let V be a real Banach space, K Assume f : K -+ R is a linearfunctional satishing

c

V be a subspace.

Then f can be extended to a continuous linearfunctional on V with

There is a related geometric form of the Hahn-Banach theorem on separation of convex sets, Proposition 2.15. For this purpose, we introduce the following definition.

DEFINITION 2 . 1 4 Let V be a real normed space, A, B c V be non-empty. The sets A and B are separated ifthere exist k' E V * ,k' # 0 and a E R such that q ~I )a 5 e ( ~ ) v u E A, E B. The separation is strict if the inequalities can be replaced by strict inequalities

51


PROPOSITION 2.15 (Separation of convex sets) Let V bea real normedspace, A , B c V be non-empty and convex. (a) Zfint(A)n B = 0 and int(A) # 0, then A and B can be separated; fiuthermore, f ( u ) < a V u E int( A ) . (b) I f A f l B = 8 and either A and B are open or A is closed and B is compact, then A and B can be strictly separated. LEMMA2.16 Let V be a real normed space, f : V Suppose -cc < a < f ( u )

-+

be convex and 1.s.c.

for some u E d o m ( f ) (hence it is possible f ( u ) = co). Then 3 ( u * ,a ) E V* x R such that In particulal; iff ( u ) # i m , then f ( v ) > a + ( u * , v - U ) V v E V, f ( v ) > -m. Proof. Every x* E (V x R)* has the form

+

( z * ,( v ,b ) ) = ( w * , v ) a*b V ( v ,b) E V x R, where w* E V * ,a* E R . Iff = co, then we choose u* = 0. Now assume f $ co. Then epi(f ) is convex, closed and non-empty. We have ( u ,a ) $! epi(f ), and the set { ( u ,a ) ) is convex and compact. By Proposition 2.15 (b), the sets { ( u ,a ) ) and epi(f) can be strictly separated in V x R.So 3 x* = ( w * ,a*) E (V x R ) * and p E R such that

+

+

( w * , ~ ) a*a > p > ( w * , v ) a*b V ( v ,b) E epi(f). Now for v E dom(f) with f ( v ) > - m , ( v ,f ( v ) )E epi(f). Hence Suppose a* 2 0. Then since u E dom(f ) , there is a sequence {v,) with v, + u as n + co. Then

+

( w * , u ) a*a > /? contradicting a

( w * , v )+ a * f ( u ) ,

( u ) .Thus we must have a*

< 0 and therefore

Then the inequality is valid for any v E V with f ( v ) > -m. A consequence of Lemma 2.16 is the following.

rn

COROLLARY 2 . 1 7 Let V be a normed space, j : V -+ be propel; convex and 1.s.c. Assume u E dom ( j ) . Then there exist u* E V * and a E R such that


Figure 2.1. Continuity of a convex function

2.3.

CONTINUITY AND DIFFERENTIABILITY

The purpose of this section is to list a few results on the continuity and differentiability of convex functions so that readers with little background on convex analysis can get familiar with basic properties of convex functions.

Continuity. The basic result concerning the continuity of convex functions is the following.

PROPOSITION 2.18 Let V be a real normed space, f : V -i be convex. (a) Assume f (u)E R. Then f is continuous at u i f f is bounded from above in a neighborhood of u. (b) I f f ifJinite on an open set M C V and is continuous at some point of M , then f is continuous on M . Figure 2.1 shows a convex function f that is continuous in the interior of its effective domain, and is not continuous at b ( f (x) = cc for x 2 b), a boundary point of the effective domain. Note that f is not bounded from above to the right of b. The next two results can be deduced from Proposition 2.18.

COROLLARY 2.19 Let M C Rd be an open convex set. Then every convex function f : M -i R is continuous. COROLLARY 2.20 Let V be a real Banach space, M c V be closed and convex. Let f : M -+ R be convex and 1.s.c. Then f i f continuous on int ( M ) . Subdifferential. The notion of subdifferential is useful in describing various mechanical laws arising in contact problems, plasticity, etc. Although in later chapters, we do not explicitly use the notion of subdifferential in deriving a posteriori error estimates, it is an important concept in convex analysis. Now we introduce the definition of subdifferential and subgradient.


Figure 2.2. Subdifferential of the absolute value function

DEFINITION2.21 Let V be a real normed space with the dual V*, and f : V -+ @. Let u E V be such that f ( u ) # +oo. Then the subdifferential off at u is defined to be the set

Any u* E df ( u )is called a subgradient off at u. We see that if d f ( u ) # 0, then f (v)

> -cc

for any v E V.

EXAMPLE2.22 For a real-valued real-variable function f : R -+ R, its subdifferential at u E R is the set of the slopes of straight lines passing through the point ( u ,f ( u ) )and lying below the curve of f . For example, the absolute value function f ( u ) = lul is not differentiable at u = 0, but is subdifferentiable there, and d f (0) = [- 1 , 11 (Figure 2.2). On the other hand, differentiable functions may not be subdifferentiable. For instance, the smooth function f ( u ) = u3 is not subdifferentiable at u = 0. The notion of the subdifferential is most suitable for convex functions. EXAMPLE2.23 (Support functional) Let V be a real normed space, K be a convex set. Consider the subdifferential of the indicator function 0 +cc

i f v ~ K , i f v $! K.

If u $! K , then d I K ( u )= 0. Assume u E K. Then u* E d I K ( u )iff

Thus we have the characterization

cV

54


Any subgradient u* E d I K (u) is called a support functional to K at u. We always have 0 E d I K ( u ) for u E K. It is easily seen that if u E int(K), then d I K ( u ) = (0). For a boundary point u E d K and the case int(K) # 0, by separating u and int(K), we can show the existence of a nonzero subgradient u* E d I K ( u ) If K is a subspace, then

which can be viewed as the orthogonal complement of K. The following important result plays a central role in the duality theory.

THEOREM 2.24 Assume V is a rejlexive Banach space, f : (-m, m ] is convexand1.s.c. Thenv* E d f ( v ) i r v € df*(v*). As is commented in Example 2.22, the notion of the subdifferential is mainly applied to convex functions. This is supported by the next result on the existence of subgradients.

THEOREM 2 . 2 5 Let V be a real normed space, f : V (a) For any v E V , d f (v) is convex and weak* closed. (b) Iff isjnite and continuous at v, then d f (v) # 0.

+

be convex.

Ordinary differentiation rules in calculus carry over to subdifferentials, either straightforwardly or with some additional assumptions. For example, it is easy to verify the following relations from the definition of the subdifferential.

A natural question is when the equality holds for the summation rule.

PROPOSITION 2.26 Let V be a real normed space, f i : V + ( - m , m ] be convex for i = 1 , . . . , n. Assume there is a uo E V such that fi(uo) E R, 1 5 i 5 n, and fi,1 5 i n - 1, are continuous at uo. Then

$'(O).Then

f ( v ) - f ( u ) 2 ( u * , v - U ) 'dv E Let v = u

+ t h, h E V, and let t -+ O+

V.

to obtain

Therefore, f ' ( u )= u*. This completes a proof of part (a).

56


A proof of part (b) can be found in [49]. Characterization of convex functions. We can use the GBteaux derivative to characterize the convexity of a function. Let V be a normed space and f : V -+ R be GBteaux differentiable. Then the following three statements are equivalent. (a) f is convex. (b) 2 f(u)+ (fl(u),v- u) v u , v E V . (c) (fl(v)- fl(u),v-u)2 0 V u , v E V .

f(4

2.4.

CONVEX OPTIMIZATION

Given a space V, its subset K, and a functional f : K -+ R, we consider the problem inf f (v). vEK

When K is unbounded, we say the function f is coercive on K if

f (v)-+ cc as iivll -+ a,v

E

K.

We have a standard general result on the existence of a minimizer to the problem (2.2).

THEOREM 2.30 Assume V is a reflexive Banach space, K c V convex and closed, and f : K -+ E% is convex and k c . If either (a) K is bounded or (b) f is coercive on K , then the minimization problem (2.2)has a solution. Moreovel; iff is strictly convex on K , then a solution of the minimization problem (2.2)is unique. This theorem will be applied later to show the existence and uniqueness of weak solutions to some nonlinear boundary value problems that are equivalent certain convex minimization problems. From the definition of subdifferential, immediately we get an extremal principle.

PROPOSITION 2.31 Let V be a real normed space, f : V -+ Then u is a solution of infvEvf (v)if0 E d f (u).

be propel:

THEOREM 2.32 Suppose V is a real normed space, K C V is a non-empty convex set. Assume f : K -+ R is convex. Thenfor u E K to be a solution of the problem inf f (v) vEK

57


a necessary and sufJicient condition is that u E V is a solution of the unconstrained optimization problem

or u E V satisfies the relation

3 u* E V * with (u*,v - u ) 2 0 'v'v E K , such that f ( v ) 2f ( u ) + ( u * , v - u ) Y v E V . The solution set is convex. I f f is 1.s.c. and K is closed, then the solution set is closed. Every local minimum o f f is also a global minimum. A minimizer o f f is unique i f f is strictly convex.

2.5.

CONJUGATE FUNCTIONALS

The idea of the duality theory can be described as follows: Let f be a given function on a normed space V . For a minimization problem

inf f ( v ),

vEV

we look for a maximization problem

such that inf f ( v )= supg(q).

VEV

~ E Q

The problem (2.3) is called the primal problem, and (2.4) is called the dual problem. Then we have the following two-sided bounds for the optimal value:

This is the basis for deriving most of the a posteriori error estimates in this book. The space Q and the function g in the dual problem (2.4) are to be constructed from the primal problem (2.3). In particular, the construction of g is related to the concept of conjugate functionals.

D E F I N I T I O2.33 N Assume V is a normed space, and let f conjugate functional f * : V * + is dejined by the formula f * ( v * ) = S U P [ ( V * ,v ) - f ( v ) ] vE V

v v* E

V*

:

V

+ E.

The

58


Using the effective domain of the functional, we can also write f * ( v * )=

sup [ ( v *v, ) - f ( v ) ] Vv* E V * . v€dom(f)

It follows from the definition that the conjugate functional f * is convex and 1.s.c. on V * . We have the following generalized Young inequality:

f * ( v * )+ f (4 2 ( v * , v ) , f * ( v * ) +f ( v ) = ( v * , v ) iff v * ~ d f ( v )

+

for all v E V , v* E V * as long as the expression f * ( v * ) f ( v )is meaningful, i.e., not of the form cc - w . This inequality is a generalization of the usual Young inequality:

where p > 1 and p* is the conjugate exponent, defined through the equality llp* l l p = 1. We will frequently need to calculate the conjugate functional for a functional defined by an integral of the form

+

Before stating a theorem on how to calculate its conjugate function, we introduce the following notion.

DEFINITION 2.34 Let R be an open set of Rd, g : 0 x lR1 -+ R. We say g is a CarathCodory function if (a) 'i 6 E R1, x H g ( x ,5 ) is a measurable function; (b) for a.e. x E R, F+ g ( x ,6 ) is a continuous function.

0 a small number. Use r = 1x 1. Let the coefficient matrix be

We consider the Dirichlet problem with f (x) = -4, g(x) = 1. Then the solution of the problem is

whereas the solution of the idealized problem is

Hence.

i.e., the exact error blows up as S + O+.

1

REMARK3 . 3 Some error estimates derived by the duality approach in this work can be shown by more elementary argument. This is the case with the estimate (3.22). Let us prove it directly. Subtracting (3.12) from (3.11), we obtain the relation


80

In particular, with v = u

-

uo, we have

Consider the quantity

Using (3.29), we have

S,

V ( u- U O ) ~ A V ( U - uo)dx

V ( u- U ~ ) ~ A V ( U uo)dx

.

[l

I

li2

V U ~ ( A ~ A-- I' ) A (A-'Ao

112

-

I ) Vuodx]

,

which is (3.22). Although direct calculations, whenever possible, are more straightforward, we note that the duality approach is a more powerful, more general way to systematically derive error estimates. In the case of error analysis for idealizations of the coefficient functions here, it is not clear how to directly prove the estimate (3.28), which is preferable to (3.22) in certain situations (cf. the numerical examples later). On the other hand, an estimate such as (3.28) naturally follows with the duality approach. In the more complicated problem setting when nonlinear boundary value problems are involved, direct derivation of error estimates appears difficult. Nevertheless, the duality approach can still 1 be employed in a systematic way. See Chapter 4 for detail.

REMARK 3.4 In the finite element method for solving boundary value problems, a time-consuming part is the construction of the stiffness matrix, that is usually done with the use of sufficiently accurate numerical quadratures to compute integrals. Thus, the actual stiffness matrix used in the finite element system is only an approximation of the true stiffness matrix. The error estimates shown in this subsection can be extended to their discrete analogues in assessing the effect of the stiffness matrix approximation on accuracy of the 1 finite element solutions.

A Posteriori Error Analysis for Idealizations in Linear Problems

3.1.3

81

AN ITERATIVE PROCEDURE

+

We consider a situation where A ( x ) = A. (x) E ( x )is known with E ( x ) = small and such that the boundary value problem with the coefficient (eij matrix A. ( x ) can be solved efficiently (say, via a fast solution solver). In this situation, it is advantageous to solve the boundary value problem with the coefficient matrix A ( x ) by an iterative procedure solving a sequence of boundary value problems with the coefficient matrix A. ( x ). Another situation where such an iterative procedure is of interest is when we need to solve a family of boundary value problems for each of them the coefficient matrix A ( x ) is close to a fixed matrix A. ( x ) . Once more, let us focus on the Dirichlet problem. We write

We assume e i j ( x )is "close" to 0. By Theorem 3.1, we have an estimate for the difference ( u - uo). It may turn out that the estimated error is not small enough. In this case, let us note that w = u - uo satisfies

Thus, if we denote wo the solution of the problem

+

we may expect that (uo wo)is a better approximation to u. Applying Theorem 3.1 to these problems, we have the next result.

T H E O R E3.5 M There holds the estimate ~ ( V U -

+

+

V ( u o W O ) ) ~( VAU - V ( U O W O ) )dx

(VWTAVW~ - q * T ~ - l q * dx ) for any q* E QE,o,where the constraint set

82


Selection of a good q* can be discussed as in Subsection 3.1.2. If the estimated error for u - ( u o w o ) is still not small enough, we may repeat the above procedure. We define a sequence { w k ) k-> lby

+

wl, = 0 For Ic

> 1, let

Then we have the following result.

T H E O R E3.6 M There holds the estimate

It is possible to provide sufficient conditions that guarantee the convergence of the iterative procedure. For this purpose, we introduce three differential operators A, A o , £ : V -+ V*, with V = H ; ( R ) and V* = H - ' ( a ) , by the formulas

+

Then, A = A. £, u = A-I k = 0 , 1 , . . . . Denote

f, uo

=

A, 1 f , and wk

= ( - A ~ ~ £ ) ~ + ~ U ~

where Ami,(Ao ( x ) ) > 0 is the smallest eigenvalue of A. ( x ) ,A m a X ( E ( x )is) the eigenvalue of E ( x ) with the largest magnitude.

83


Consider the operator A;'& : V -+ V, and let us bound its norm. Let w = &'&v for any v E V. We have Aow = &v, i.e.,

Taking x = w in the relation, we obtain collwll$ 5 eollwllvllvllv,

Now if E ( x ) is small in the sense that eo < co, then by the geometric series theorem (see, e.g. [6, page 46]), Ego(-A,'&) converges to (I A;'E)-~. Then, as k -+ oo,

+

converges to

Meanwhile, wk -+ 0 in V as k q* = -EVwk-1 gives:

-+ oo.

The error bound in Theorem 3.6 with

A simple error bound is obtained by dropping the second integrand from the integrand:

84


which is in turn bounded by

with An advantage of using the iterative procedure is that if the finite element method is used to approximate uo, wo,wl,. . . , then the stiffness matrices are the same for all the finite element systems. This provides substantial time saving in forming and solving the finite element systems. The above consideration works for other kind of boundary value problems as well.

3.1.4

NUMERICAL EXPERIMENTS

We present here several numerical examples to show the effectiveness of the error estimates derived in this section.

EXAMPLE 3.7 Consider the problem

and its idealization

Here, R is a two-dimensional domain, aij ( x ) = dij Kronecker delta symbol:

+ eij ( x ) ,and Sij is the

Let E ( x ) = ( e i j ( x )be ) symmetric (to ensure the symmetry of the coefficient , we matrix A ( x ) ) . We assume some knowledge about the size of e i j ( x ) but do not assume these functions are known exactly. Suppose first that we know some bound on the eigenvalues X I ( x )and X2(x)of E ( x ) :

for some E E (0,1 ) . We use the selection (3.21) to get the estimate

V ( U- U

O ) ~ VA( U -

u O dx ) 5

S,

+

VU;(A A-'

-

21)Vu0dx.


85

To proceed further, we have

Then

and so

By using the bounds on the eigenvalues of E(x),we have

where the matrix norm

11 . 11 is the spectral norm. Thus, we have the estimate

For example, if the domain is the unit square S2 = (0,I ) ~ ,f (a) = 1,

g(x) = 0,then / / V U ~ / /=~ 0.1874679.. Z(~) ..

As another example, we choose S2 to be an L-shape domain,

f (x)= 1, g(x) = 0. Then

86


We note that the estimate (3.34) is optimal. For if we assume eij ( x ) = -edij, then u = uo/ (1 - E). Thus,

More often, for the perturbation matrix E ( x ) ,we are given a bound on its entries rather than that on its eigenvalues. We assume

where

E

E

(0,O.s).Then, the estimate is

EXAMPLE 3.8 We reconsider the boundary value problem (3.30)-(3.3 1) and its idealization (3.32)-(3.33). This time, the coefficient matrix is assumed to be

where r = 1x1. We assume the coefficient function has the property that a ( r ) -+ 1 as r -+ I , and a ( r ) -+ 0 as r -+ 0. This property is assumed so that the differential equation exhibits singularity near r = 0. As examples of the coefficient function, we may take

The domain is a circular comer region with an internal angle w :

0 = { ( r , 8 ): 0 < r < I , 0

< 8 < w).

Denote a = T / W . Let f and g be such that the solution of the idealized problem (3.32)-(3.33) is ug = ra sin a8. We seek the solution in the form


It turns out that R is a bounded solution of the problem

In particular, for

a ( r ) = r'ln, n > 1 ,

Then it can be shown that the exact energy norm error is

The estimated error bound is

where q* E ( L( ~R ) )satisfies ~ the condition

We select q* according to the formula (3.25). For our problem,

V ( r ,0 ) = a r y - y - l r y + l

li2

sin a0 d0

After a tedious calculation, we find an analytic expression of the error bound:

where we assumed 2 ( y

+ 1 ) > lln. Note that a2 min 2(y+l)>l/n2 ( y

+ ( y+ I ) ~ +1) - l/n

88


For this y,the second term of (3.35) is negative, so we take

In conclusion,

To see the efficiency of the error bound, we use the effectivity index

Some numerical results are given in Table 3.1 on the values of the true error, error bound, and the effectivity index, for several choices of the internal angle w and the parameter n. We observe that the efficiency of the error bound is rather stable with respect to the increasing strength of comer singularities, i.e., as the singularity strength increases due to an increase of the internal angle, the effectivity index does not grow out of control. We also observe that the efficiency of the error estimate improves as the parameter n increases.

EXAMPLE 3.9 We again consider the problem (3.30)-(3.31) and its idealization (3.32)-(3.33) with the assumptions stated in Example 3.7. Consider the iterative procedure described in Subsection 3.1.3. Applying Theorem 3.5 with q* = -Vwo, we get the estimate

+

(VU - V(UO

W O ) ) ~ A( V u -

+

V(UO

WO))

dx

We then obtain

We can bound IIVwo11 L 2 ( R ) in terms of IIVuo11 L2(n),using the definition of Wo :


Table 3.1. Example 3.8, effectivity of error bound

i.e.,

0 is the smallest

> 0 is the smallest

For general domains Rl or general decomposition of d R 1 into 71 and 72, we have to compute X1 numerically. However, for certain special situations, it is possible to find XI analytically. In the following examples, we consider Dirichlet problems.

EXAMPLE 3.11 Assume R1 = (0,a)x (0,b) is a rectangle. For convenience, in this example, we use (x,Y) instead of (xl, 2 2 ) to denote the coordinates of a


95

point in 2.We use the method of separation of variables to solve the eigenvalue problem (3.52)-(3.54). In our special case, yl = ( ( 0 ,a ) x ( 0 ) )U ( ( 0 ) x ( 0 , b ) ) , "/z = ( ( 0 ,a ) x { b ) ) U ( { a ) x ( 0 ,b ) ). Substituting the expression v ( x ,y ) = X ( x ) Y ( y ) into (3.52), we find two sub-eigenvalue problems

XI1 = - ( A - p ) X in ( 0 ,a ) , X ( 0 ) = 0 , X 1 ( a )= 0 ,

(3.58) (3.59)

and

Y" = - p Y in ( 0 , b ) , Y ( 0 ) = 0 , Y 1 ( b )= 0 . Solving (3.60)-(3.6 I ) , we get a sequence of eigenvalues

and a sequence of corresponding eigenfunctions:

For each pl, the eigenvalue problem (3.52)-(3.54) admits a sequence of eigen-

with a sequence of corresponding eigenfunctions:

Thus, for the eigenvalue problem (3.52)-(3.54), we obtain a sequence of eigenvalues:

with a sequence of corresponding eigenfunctions:

It is known that (cf. [141]),{Xk)F=oforms an orthogonal basis of the space {V E H 1 ( 0 ,a ) : v ( 0 ) = 0 ) , { ~ ) forms ~ oan orthogonal basis of the space

96


{ v E ~ ' ( 0b ), : v ( 0 ) = 0 ) . Thus, the sequence (3.63) forms an orthogonal basis of the space { v E ~ ~ ( 0 : v1 = ) 0 on 7 1 ) . Therefore, all the eigenvalues of the problem (3.52)-(3.54) are X k , ~k, , 1 = 0 , 1 , . . . , with the corresponding eigenfunctions X k ( x ) x ( y ) ,k , 1 = 0 , 1 , . . . . In particular, the smallest eigenvalue is

The estimation in this special case is

E X A M P L E3.12 Assume Rl is a circular corner domain: 0 1 =

{ ( r , Q ): 0

< r < r0,O < Q < w ) .

In this case, yl = { ( r , Q ): O < r < r o , Q = O , w ) , yz = { ( r , ~: )r = ro,O < Q < w ) .

By expanding v into the series C R k ( r )sin k a Q , a = T / W ,we see that the eigenvalue problem (3.52)-(3.54) is reduced to the problem: For each positive integer k , find the eigenvalues X of

lTO < R: ( r )r dr

co.

On the other hand, by using (3.49), we obtain

X 1 = min Therefore, X 1

1

J; [ r ~ ' ( r+)~~ ~ R ( r ) dr ~ / r ] :J

~R(r)~dr

> 0 is the smallest number such that the problem


has a nontrivial solution R = R ( r ) . By making the change of variable

we conclude that XI

> 0 is the smallest number such that the problem

has a nontrivial solution R = R ( r ) . From the differential equation (3.64) and the condition (3.66), we find that

where J, is the Bessel function of the first kind of order a (cf. [104]),

An integral representation of the Bessel function is Ja(4 =

(~/2)"

1

Thus by (3.65), XI

+1

In

cos(r cos 0) sin2a do.

(3.67)

0

> 0 is the smallest root of the equation

Using the representation formula (3.67), we see that 6ro > 0 is the smallest root of the equation

la

sin2, B (or cos(r cos 0) - r cos 0 sin(r cos 0)) d0 = 0.

(3.68)

We use Newton's method to solve (3.68). The algorithm is the following. (a) Initialization. Find an r ( ~>) 0 close to the smallest root of the equation (3.68). (b) Iteration. For k = 0,1, . . .,

where, c(k) =

J:

J: sin2ff0 (a cos(ck(0)) - r ( k )cos 0 sin(ck(0))) d0 0 ((1 + a ) cos 0 sin ck(0) + dk)cos2 0 cos ck (0)) de

98


Table 3.3. Example 3.12, smallest roots for several angles

and ck (8) = d k )cos 8. For the computation of the two integrals on the right hand side of the iteration formula, we divide the integration interval into m equal-length subintervals, and on each subinterval, we use the six point Gauss quadrature formula. Numerical H results for several values of the angle are listed in Table 3.3.

3.2.2


Below, we present some numerical results to show the effectivity of the error estimate (3.50).

EXAMPLE 3 . 1 3 Assume R = (0, I ) ~E , = 0.2, R, = ( 0 , 0 . 2 ) ~and ,

whereas the error bound from the estimate (3.50) is

If, instead of (3.69), we assume

Then,

jlV(u,

-~

= 0.006141 .

) l l ~ 2 ( ~ )


'

(3.72)


99

Comparing (3.70) with (3.71), and (3.72) with (3.73), we see that our estiW mates provide good error bounds. Let us examine another example.

EXAMPLE 3.14 Assume 0 is an L-shape domain:

and

We choose

E

= 0.3. Then

From Example 3.12, we have 6= 1.4013. - - (note that ro = I). Thus, the estimate (3.50) implies

If, instead of (3.74), we assume

then

IIV(u, - u ) I I ~ ~=( 0.01256 ~) -

' '


Now assume in fl\fl,, Here as usual, we use r = 1 x 1. Then


In all the cases, we see that our estimates provide good numerical results. W


100

3.3.

BOUNDARY CONDITION IDEALIZATIONS

The boundary condition idealization is one of the most delicate aspect of the reliability of the computational analysis. Let us mention a concrete example which will be considered in detail in Section 3.6. Consider a heat (or membrane) problem leading to the Poisson problem with zero boundary condition. In reality, such a boundary condition does not exist, because the temperature (or displacements) can not be completely controlled by the surrounding medium. A more accurate description of the boundary condition would be of the type c ( x ) d u l d u u = 0.The function c ( x )has small values and depends globally on many factors. In practice, usually a set of boundary conditions needs to be analyzed. For some discussions in the context of linearized elasticity, we refer to [7], where the influence of boundary conditions on the reliability of the results is addressed in a computational way. Let us mention that the uncertainty in the boundary condition is especially large in the neighborhood of corners of the domain, where the theoretical fluxes (or stresses) could be infinite. A typical example in the elasticity is the plate paradox for hard, soft supports which practically can not occur in reality, because such conditions can not be practically imposed. For detail, see e.g., [9]. Hence, estimates of boundary condition idealizations are of major importance. In this section we show how to employ the duality technique to derive a posteriori error estimates for boundary condition idealizations. We will consider two idealization problems. Slight extensions of the results will be used in quantitative error estimation of some idealization problems related to heat conduction in Subsections 3.6.2 and 3.6.3.

+

BOUNDARY CONDITION IDEALIZATION (I) Let fl c Titd be a Lipschitz domain with its boundary dfl = Fl U G, rl relatively closed, r2relatively open, rl n r2= 0,and meas(rs) > 0.Let there 3.3.1

be given f E L2( Q ) , g E H1( a ) , A E Lw ( r 2 ) such that for some constants 0 < A. < Al < m,

We will be interested in the case where A. is of large size. Consider the boundary value problems


101

and -

nuo = f uo = g

infl, ondfl.

Note that on F l , the two problems have the same Dirichlet boundary condition. On F2, since the coefficient function A ( z ) is assumed big, the condition (3.77) is close to the condition (3.79). The weak formulation for the boundary value problem (3.75)-(3.77) is to find u E g H& (0)such that

+

and that for the boundary value problem (3.78)-(3.79) is to find uo E g+ H i (fl) such that S,vuo.vvd~= f ~ VdU E ~H ~ ( R ) .

S,

Our purpose here is to bound the difference (u - uo). We introduce the folowing function spaces:

and define the operator A E C(V, Q ) by

Let J(v,Av)=F(v)+G(Av),

VEV,

with

Here for q E Q, we write q = ( q l , q2) with q l E (L2(0))d , q 2 E L2(F2). A similar notation is used for q* E Q*. Then u E V is the unique solution of the minimization problem inf J ( v , Av).

vEV


102

For a lower bound of D ( u ,u o )defined in (2.12), we note that

Hence, by (2.21),

D(u, u 0 ) =

lo(1

+ t(AuO- Au)) (AuO- Au),AuO - Au) dt

- t ) (Gtt(Au

We now compute the conjugate functions.

F* (A*q*)= sup{ (Av,q*) - ~ ( v ) ) v€V

Introduce the constraint set

Then we have

For the conjugate function of G , we have


Thus,

Hence, applying Theorem 2.40, noting uo = g on we have the next result.

r2,and rearranging terms,

T H E O R E3.15 M For the difference between the solutions of (3.75)-(3.77) and (3.78)-(3.79), there holds the estimate:

for any q* E QE. Assume d u o / d v E L 2 ( r 2 ) .Then from (3.78)-(3.79), we have

*

So a simple choice of the auxiliary function is q; = -Vuo, q2 = and we obtain from Theorem 3.15 the following estimate:

Using the bound A ( x )

auo a V

-

A g,

> Ao, we obtain from (3.80) that

In the derivation of the estimates (3.80) and (3.81), we assumed d u o / d v E L 2 ( r 2 )on the solution uo of the problem (3.78)-(3.79). This assumption is valid if the domain is smooth, or the domain is a comer domain without cracks, and g is the trace of an H~ (R) function.

104

3.3.2


BOUNDARY CONDITION IDEALIZATION (11)

Let there be given a function a on dR with where, a0 I a1 are two positive constants, supposed to be small. Also given are f E L2( R ) ,g E H1( R ) ,and h E ~ ~ ( 8 0We) assume . P

Consider the boundary value problems

and

respectively. A weak formulation of the boundary value problem for u is to find u E H 1 ( R )such that

The compatibility condition (3.83) is assumed to ensure the idealized problem (3.86)-(3.87) has a solution, which is unique up to an additive constant. We use the duality theory to estimate ( u - u o ) .Let

d

For q E Q, we write q = ( q l ,qz) with ql E ( L ~ ( R ) and ) q2 E L2( d R ) . A similar notation is used for q* E Q*. Define the energy functional

J ( v ,Av) = F ( v ) with P

F ( v ) = - Jn f v d x ,

+ G(Av)


Then u minimizes the functional J ( v , Av) in V. For the Gsteaux derivatives of G, we have

Thus, from (2.21), we can find

We compute the conjugate functions. Define a constraint set

The constraint condition is a weak form of

We then have

and

Thus, J*(A*q*,-a*) =

{

+

1

(ah q 2 an 2a +m otherwise.

+g -)

ifq* E

Q:,

Hence, applying Theorem 2.40, we have the following error estimate.

THEOREM 3.16 Forsolutionsof (3.84)-(3.85) and (3.86)-(3.87), there holds the error estimate:

106


for any q* E Q*.

In particular, by the weak formulation of the boundary value problem (3.86)(3.87), we see that q* = ( - V u o , g ) is admissible. Then we have the error estimate

Using the smallness assumption (3.82), we have

for any solution uo (which is unique up to a constant) of the problem (3.86)(3.87).

3.4.

DOMAIN IDEALIZATIONS

Domain idealization can be a source of many difficulties in applications. As an example, in linearized elasticity, domains with reentrant comers lead to infinite stresses and strains, violating basic assumptions in deriving the differential equations in the theory of linearized elasticity. Moreover, except for cracks, it is impossible to fabricate exactly a comer. Hence, a comer has to be understood as a mathematical idealization with possibly non-physical consequences for some data of interest. Also, the domain often comes from a process of optimal design. It is important to have an approach for quantitative estimation of the effect of domain idealizations, especially in the neighborhood of comers, edges, etc. In this section, we consider two types of domain idealizations, for one of which, the problem domain is included in the idealized domain, whereas for the other, the problem domain includes the idealized domain.

3.4.1

DOMAIN IDEALIZATION (I) Let R, c R be a subset of R around a comer 0. Assume f

E L ~ ( Rand ) g E H 1 ( R ) are given. We apply the duality theory to give an a posteriori error estimate between the solutions u, E H1(R\R,) and uo E H 1 ( R )of the boundary value problems

and


We set

v = H~(R\R,),

V* = ( H I (R\R,))*, Q = Q* = ( L 2 ( f l \ W 2 , A = V : the gradient operator, J ( v , Av) = F ( v ) G(Av),

+

where

f v dx if v E V, v = g on aR\dR,, otherwise,

Then u, E V is the unique solution of the minimization problem inf J ( v , Av).

vEV

From Example 2.4 1, we have

Let us compute the conjugate functions. Introduce the constraint set

The constraint condition is a weak form of the relations divq* = f in R\R,, Then *

*

* =

{

"a; +oo

q * . v = 0 on dR,\dR.

( q * . V g + f g ) d x i f q * E Qz, otherwise

A POSTERIORI ERROR ANALYSlS VIA DUALITY THEORY

108

Thus, we obtain the following general error estimate

for any q* E QE. Now we discuss how to choose a suitable q* in (3.88). We will use the same notations for Ro,,, , Rr,, Fr1, vl.and so on as in Subsection 3.1.2. We assume there exist two positive numbers rz < r l , such that R, c

Define

where 6 > 0 is a constant to be suitably chosen. Then we take

It can be verified that the above q* is an admissible function, i.e., q* E QE. Using this selection q * , the estimate (3.88) becomes

3.4.2

DOMAIN IDEALIZATION (It)

Let R c Ktd be a Lipschitz domain, and 0,a small-size Lipschitz domain neighboring R (Figure 3.3). Let RE be the combined domain such that EE = RU Assume dR = U with n r2 = 0, d o E = U with

n,.

rl n r 2 , =~ 0.


Figure 3.3. Domain idealization

Let uo E H 1( R ) ,u, E H 1 ( R E )solve

and

We make the following assumptions:

and

A POSTERIORI ERROR ANALYSIS VIA DUALlTY THEORY

110

Let us derive an estimate for the difference between u,and uo.Take

Let J ( v , Av) = F ( v )

+ G(Av),

where F(v) =

{ '.Efvdx-12,E g2q2

+m

if ?I E a f H;, (OE), otherwise,

Av) among all functions v E gl Then u,minimizes J(v, For the functional G, we have

+ H;,

(RE).

Let Go E H1(RE) be an extension of uo to QE. Then from (2.21) for the we have quantity D ( u ,Go),

Now we compute the conjugate functions. Define a constraint set

The constraint is a weak form of the relations divq* = f in OE,

-q* . Y = 92 on r 2 , ~ .

Then we have F*(A*q*)=

(4' . Vgl

+ f g i ) dx +

/

~ 2 ~ 1if dq*~E

Qf,

r2,~

otherwise


and

Thus,

/q*I2+ q f ' V g l J * ( A * q * ,- q * ) =

+cc

+ f gl

if q* E Q z , otherwise.

Using Theorem 2.40, we obtain the following estimate 1 2 -I W - C0)/IL2(ilE) 2

for all q* E Q z . Recall the assumption (3.95). We can choose q* =

{ ,vuo

in,, in R,.

Noting the relation

we then have from (3.96) the following estimate

I/~(U

-CO)IIL~(~ I ~llvfioll~2(n~)~ E)

E X A M P L E3.17 Assume R is a corner domain with an internal angle w < 2 7r at the corner 0.Assume uo = X r f f cos a 0 is the solution of the problem (3.89)-(3.91) with homogeneous Neumann boundary condition around 0, where a = 7rlw. Assume R,

c { ( r ,0 ) : w < 0 < 27r).

Then, a simple choice of Go in R, is

1 uo = -r a ( 2 0 - 27r - w ) . 277 - w For this choice of Go, the estimate is 1 l l -~ C O I L ~ ( ~ 0, p E ( 0 , l ) .

The weak formulation of the boundary value problem is: Find u E H;l such that

,gl

(0)

We then introduce the corresponding minimization problem. Let

v = H ~ ( R ) ,v * =( H ~ ( R ) ) * , d

Q = Q* = ( ~ ~ ( ,0 ) ) Av = Vv. We define the energy function on V: J ( v , Av) = F ( v )

+ G(Av),

where F(v) = otherwise,

Then the weak formulation (4.9) is equivalent to the minimization problem inf J ( v , Av).

vEV

We can apply Theorem 2.37 to show that the minization problem (4. lo), and hence the weak formulation (4.9), has a unique solution u. We call Q(u) = {x E 0 : [ V U ( X ) ~to)

T. Let a = T/W. When on both sides of the part of the boundary around 0,either a Dirichlet or a Neumann boundary condition is specified, we have

lVuo( x )1

N

O(ra-I) for r = Ix / close to 0.

When on one side a Dirichlet condition is specified, while on the other side a Neumann condition is specified, we have

IVuo( a )1

N

O ( T " / ~ - ~for ) r = Ix I close to 0.

Since a < 1, l V u o ( x ) /-+ oo as x -+ 0. Hence, it is doubtful whether the linerized problem (4.11)-(4.13) is in any sense close to the nonlinear problem (4.5)-(4.7). We will derive an estimate for the difference between u and uo. Why the coefficient function a ( ( ) is chosen in the form (4.8)? The onedimensional stress-strain relation corresponding to the nonlinear problem (4.5)(4.7) is a = ~ ( 1 ~ E1 . ~ For ) scalars a and E , we require a stress-strain relation as illustrated in Figure 4.1, where we omit the part of the graph for negative a and E, which can be obtained by reflecting the graph for positive a and E with respect to the origin. The coefficient function a ( < )corresponding to Figure 4.1 is of the form (4.8) with ,B = E T / E , to= E $ . Now suppose we have computed the solution uo of the linearized problem (4.11)-(4.13). We are interested in bounding the error ( u - uo)by using uo.

A Posteriori Error Analysis for Linearizations

Figure 4.1. One-dimensional stress-strain relation

Let us first compute the conjugate functions. F*(A*q*)= sup [(Av,q * ) - F ( v ) ] vEV

=

sup

[S,(q*."v+fv)dx+

V E H ~(a) ~

L r

dsI


The constraint condition is a weak form of the relations divq* = f i n R ,

-q*v=gzonr2.

Then F*(A*q*)=

(Q* . Vgi

+f

gi) dx

+

gzgl ds if q* E

QE,

otherwise.

132


We also have

=

/

f?

=

q -

sup

,ad

y

I

*t -

1lq"

l2$0

:a(() dt] dx

dt] dx

Here, t = b(/q*1 ) is the solution of the equation

We can determine that

In computing G*( - q * ) , we used Theorem 2.35, also we assumed q* Obviously, the expression for G*( -q*) holds for q* = 0 too. Thus,

# 0.


133

Now we consider the quantity D ( u , uo) defined in (2.12). We use Theorem 2.40 to derive an upper bound:

Hence,

for all q* E QE. Note that we also have the bound for the energy difference E D ( u , uo) of (2.11) between u and uo:

for all q* E QF. Then we turn to derive a lower bound for D ( u , uO).From the definition of the functional G, we can find


134 Thus, by (2.20), 1

=I

( ~ ' ( h+u t A(uo - u ) ) - G 1 ( h u )A(uo , - u ) )dt

-

a ( V u 2 )V U ] . V ( u o- u ) dxdt

- V (uo- u ) dr dx dt.

+

6 , using the definition of the coefficient function a (.) , we see that the integrand is not less than P t I V (uo- u )1 2 . Now denote

+

for the linear and nonlinear regions of uo and of both u and uo. Then on f l l ( u ,uo), jVu+t.rV(uo - u ) l
0 such that in the region Ro = R n {r > ro), JVuol5 where w is the internal angle of the comer. Denote ro = fl n {r = ro), vo the unit normal vector on roin the direction of being away from 0. We further assume

do/ d m

f (x)= 0, x We define a function in R\Ro:

E

R\flo.

,

(4.21)


Figure 4.2. Neighborhood around a comer

Then we take

in Ro, in R\Ro.

This function is admissible, since for any v E - i q * .

Vvdx =

lo

V u o. V v d x

HA ( R ) ,we have

+ l , n o ( V y ,- V z ) V v d x

The selected function q* has the property that Iq* (x)1 5 do. This inequality is obvious for x E Ro. For x E R\Ro, we use the relation

to derive the bound


Then we have the estimates -a(E) d
0 such that in Ro = R n { r > r o ) , IVuoJ5 & / d m . Again, denote ro = R n { r = ro). Besides the assumption (4.21), we also assume that g2 vanishes around 0 , i.e.,

where, instead of (4.22),

From the linear problem for uo and the assumptions made on f and g2, we have the equality

< 6.

Hence, q* is an admissible function. Once more, /q*(x)1 Therefore, the estimates (4.24) and (4.25) hold. The situation where the boundary condition around the comer changes its type can be treated similarly.

4.1.3


We present two numerical examples to show the effectiveness of the estimates (4.19), (4.20), (4.24) and (4.25). We will compare the estimates resulted from

138


the two different selections of the auxiliary function q*. For the first example, we know the exact solutions for both the nonlinear problem and the linear problem. Hence, we will see how effective our estimates are by comparing our error bounds with exact errors.

EXAMPLE 4 . 1 We take R to be a unit disk excluding a small hole

where r, > 0 is a small number. Let us consider a family of nonlinear problems with a small parameter X > 0: in R, for r = 1, for r = r*,

-div [ a ( V u 2 ) ~ u=] 0 u = X ln(l/r,) u =0

where a(.)is the function defined in (4.8). The true solution is d o r llnr

+ X In-r*1

forrl

I

r r1h--(l-P)(r-r,) r*

< r < 1,

forr* < r < r l ,

where rl is the unique positive solution of the equation

with ro = A/& The linearized problem

-nuo= 0

in R, for r = 1, for r = r*,

uo = X ln(l/r,) uo = 0

has the solution It is easily verified that rl

< ro. Therefore,

Thus, in some sense, the linearized problem is more "rigid" than the nonlinear problem.


139

We take to = 1, r, = 0.01. For various values of /? and A, we have the following numerical results. Here

is the square root of the difference in energy,

is the error in an energy-like norm. We have

For Selection 1, q* = -Vuo,

we obtain an upper bound for E J ( u ,u o )and E ( u ,u o ) :

For Selection 2,

where 7-2 = rod-, E ( u ,uo):

we have another upper bound for E j ( u ,u o ) and

140


Table 4.1. Example 4.1, true errors and error bounds, X = 0.1



to be the effectivity index of our error estimates. Numerical results for various values of the parameters X and P are reported in Tables 4.1, 4.2, and 4.3. is a very good error bound for any values of We see that min{Eestl, Eest2) ,B and A. For ,B away from 0, Eestlprovides a better bound, whereas when ,f?is close to 0, we should use Eest2 as our error bound.

EXAMPLE 4.2 Let R be a circular comer domain


141

We use again the notation cu = T / W . Consider a particular case when the linear solution is (4.30) u O ( x )= X r f fsin ae, where X

> 0 is a parameter. Since IVuo I = X a r f f - I ,the linear region of uo is

where, the radius Xa

1/(1-~)

r0 =

Let us estimate the difference between uo and u, the solution of the nonlinear problem

We try to give an upper bound for both EJ ( u ,u o )and E ( u ,u o ) ,defined in Example 4.1. Denote

If we use Selection 1, q* = -Vuo,

then an upper bound for the error ( u - uo)is

If we use Selection 2,

where

and

V ( r ,6') = -ar?-lr cos cue in R1,


Table 4.4. Example 4.2, error bounds for problems on an L-shape domain

then an upper bound for the error is

.

We take to= 1, and w = 3 ~ / 2(i.e., Q is an L-shape domain), the numerical results are given in Table 4.4. We see that indeed, EeStlprovides a better bound for P away from 0,whereas Eestzis better than Eestlwhen /3 is close to 0. We end the subsection with several remarks.

REMARK4 . 3 We have seen in the numerical examples above that our a posteriori error bounds for the continuous problems are effective. When P is not close to 0, the error bound from the simple Selection 1 is better than that from the Selection 2. On the other hand, when P is close to 0, the error bound from the Selection 2 is better than that from the Selection 1. By continuity, at least when finite element solutions are sufficient close to the exact solutions, we can claim that our a posteriori error bounds for the discrete problems are effective too. When P is not close to 0, the error bound (4.20) is better than (4.25), whereas when ,B is close to 0, the error bound (4.25) is better than (4.20).

.

REMARK4.4 In the above discussions, we assumed a concrete form for the nonlinear problem. This is only for the purpose of making specific computations and of showing the effectiveness of the error estimates. In fact, we do not depend on a concrete form for the nonlinear problem. To be able to provide error estimates for the effects of mathematical idealizations on solutions, we


143

only need a description on the structure and range of the coefficient function. This is a crucial point which allows the useful applications of the suggested a posteriori error estimates on practical problems, where all the data contain certain degree of uncertainty. Of course, the more accurate the description of the nonlinear problem, the more accurate the error estimates. REMARK4.5 In this section, we consider the effect of the linearization. The mathematical idealizations of a practical problem can be more complicated than the linearization, and the general framework for deriving a posteriori error H estimates discussed earlier allows us to get similar estimates. REMARK4.6 The sample problem discussed above is an elliptic boundary value problem for a Laplacian-like differential operator. The idea for deriving a posteriori error estimates works for more complicated problems in applications. 1 One such example is given in the next section.

4.2.

LINEARIZATION OF A NONLINEAR ELASTICITY PROBLEM

The classical theory on linearized elasticity is based on the assumption that the size of the displacement gradient tensor is small. The linear Hooke's law in the linearized elasticity theory characterizes a large class of elastic materials occupying smooth domains subject to moderate size body forces and boundary tractions. For engineering applications, however, the ideal assumption is not always satisfied. For instance, consider an elasticity problem on a domain with reentrant comers. Domains with comers and edges are common in engineering applications. The solution of the linearized elasticity system exhibits singularities; in a neighborhood of the comers, the strain and stress tensors computed from the solution become arbitrarily large in sizes. Thus, the basic assumption of the linearized elasticity theory is grossly violated around the comers. On the other hand, the linearized elasticity model is used for many applications, as long as it can be reasonably expected that the violation of the basic assumption is limited to a small region of the domain. Qualitatively, when the input data are small, the expectation is validated by the continuous dependence of the deformation of an elastic material on input data. Obviously, it is more useful to have quantitative information on the error caused by using the solution of the linear problem to describe the response of the elastic material on a comer domain. The purpose of the section is to provide quantitative error analysis for the effect of linearization on solutions of nonlinear Hencky materials. We notice that in practice there are material models more realistic than the nonlinear Hencky materials considered here. However, we may view the analysis given in this section as a preliminary step towards a general a posteriori error analysis

144


for linearization of more realistic materials; mathematical theory of nonlinear elasticity can be found in [42, 106, 1221. A typical result has the feature that we solve the linearized elasticity problem and use its solution uo to bound the error ( u - u o ) , with u the solution of a nonlinear Hencky material problem. We emphasize here that to obtain an error bound, all we need to know is the solution of the linearized elasticity problem.

4.2.1

NONLINEAR HENCKY MATERIALS

For a detailed description of nonlinear Hencky materials, one is referred to [ I l l ] or [160]. Let R c EXd be a d-dimensional domain. For a displacement field v : R + EXd, we use the notation v = (vl , . . . ,vd)Tand introduce the infinitesimal strain tensor E(V): R + S d with components

Denote by a the stress tensor with components the stress and strain tensors are defined by

aij,

1

< i, j

5 d. Deviators of

where t r ( r ) = q i , t r ( ~ = ) ~ i i Recall . that for the dot product of two tensors = (E . E ) " ~ . ~ , EqS d, E . q = E i j q i j Also, For a linearly isotropic, homogeneous material, the stored energy function is 1 Mo(E)= I X ltr(E)I2 p

+

or written as

1 Mo(E)= 2 k tr(c)12

+ p 1cD12,

+

where k = X 2 p l d is the bulk modulus, X and p are the Lam6 coefficients. A nonlinear Hencky material is characterized by a stored energy function of the form 1 (4.31) M ( E ) = 5 k ltr(c)I2 p 4 ( / E D 1 2 )

+

for a non-negative, continuously differentiable function

a then The constitutive law a = d M ( ~ ) l d is

4 satisfying


and we have uD = 2 p

$'(IE~/~)

E

~

.

For the special case when 4 ( [ ) = 5, we obtain a linearly elastic material, and the constitutive law becomes

and we have uD = 2 p s D .

Now we assume a nonlinear Hencky material, initially occupying the domain

R,is subject to the action of a body force of density f , a traction g2 on part of the boundary r2c dR.We assume on another part of the boundary rl c do,the displacement field gl is specified. We may specify other boundary conditions as well, e.g., normal or tangential components of tractions or displacements. One can easily extend the a posteriori error estimates to be presented below for the cases of other boundary conditions. Let us assume

and meas(rl)

> 0.

We denote the Sobolev spaces

Then the displacement field u : R + Rid of the nonlinear Hencky material minimizes the total potential energy:

where the total potential energy

We now study the minimization problem (4.35). We need the following result.

LEMMA 4.7If R is a Lipschitz-continuous domain or a crack domain, then E ( v ) + m as lvllv

+ m, v - g1 E Vrl.

Proof. We will use Korn's inequality (page 34) lk(V)12dX 2 co llvll2. a v E v r , .

146


Usually, Korn's inequality is proved for a Lipschitz-continuous domain ([l 1I]). For a domain with cracks, we decompose R into R1 and R2 in such a way that both R1 and R2 are Lipschitz-continuous, meas(dRl n F l ) > 0 and meas(df12 n r l ) > 0. Then for a v E Vrl, Korn's inequality holds for both vial and via,. Thus, Korn's inequality also holds for v. For v E V ,v - gl E Vr,, using the assumption (4.32) and Korn's inequality (4.36) on v - g l , we have

THEOREM 4.8 Let R be a Lipschitz domain or a crack domain. Assume (4.33) and (4.34). If we further assume 4(J) is second-order continuously differentiable and satisfies, for some a > 0,

then the minimization problem (4.35) has a unique solution u E g l solves the nonlinear elasticity system

+ Vrlthat

Proof. First we prove that the stored energy function M ( E ) defined in (4.31) is strictly convex. The second-order GBteaux derivative of M at E along the direction q is


Here, we used the assumption (4.32) and the relation

If

4 " ( l ~ ~ t o , from (4.39) and (4.48), we have

It can be bounded from below as follows:


154

Using the assumptions on the parameters

Recall that k = X

and y, we have

+ 2 pld.

where the coefficient

is a positive constant by the restrictions on ,f? and y. We can also show directly that the equation (4.43) is uniquely solvable. 2 Obviously, we only need to prove the function f (J) = ($I([)) J is strictly increasing when J > to.In that case,

and

If y

2 112, then

If y

< 112, then

Hence, f (J) is a strictly increasing function.

155


Finally, we modify the inequality (4.47), which is inadequate when the parameter ,f? is close to zero. By (2.20), we have

where h ( t , u , u o= ) 4' (1cD(u) +tcD(uO- u)12)( c D ( u+ ) tcD(u0 - u)) cD(u0- u ) - 4' (IcD(u)12) c D ( u .)cD(u0- u ) . Define the set n l ( uu , 0 )= (3: E fl : ~ E U ( U ( X )I) ~Eo,~

Eo)

I E ~ ( u o ( ~ : )I )I~

and then, h ( t , u , u O=) t (cD(uo - u)I2. On the remaining part of 0 , it can be verified that h(t,u , uo) 2 t P l c D ( w- u)l2 if y >_ 112, and

if y

< 112. These are proved in a fashion similar to the proof of (4.50). Define

Then we have the following relation:

156


In conclusion, we have the following result.

T H E O R E4.10 M For a nonlinear Hencky material defined by the governing finction (4.48),the following a posteriori error estimates for the linearization of the material hold: For the energy diference,

+

- 2 [ c D ( u o ) 3 ( - 2 ~E ~ ( u o ) ) ] . E ~ ( u o ) } d x ,

(4.51)

and also

-2

+

[ c D ( u 0 ) + ( - 2 ~~ ~ ( u o .)s D ) ]( u 0 ) }dx. (4.52)

As for numerical experiments, we consider elasticity problems on a planar comer domain R (d = 2). Near the reentrant comer 0,the domain R coincides with a cone

C, = { ( r , ~: r) > 0 , -w/2 < 0 < w / 2 ) . Assume the body is subject to boundary tractions only and in the neighborhood of the comer it is traction free. We have the following singularity expansion for the solution uo of the linear elasticity problem ([146]):

Here, Go is a smooth remainder such that &(Go)is bounded when x + 0 , G is the modulus of rigidity, K I and K I I are Mode 1 and Mode 2 stress-intensity factors,

Q I ( X I+ 1 ) ) cos X I O - X I C O S ( X I - 2)Q + QI(X1+ 1 ) )sin X I O + X I sin(X1 - 2)O

(K (K

Q1l(e)=

(

(K -(

K

+

Q I I ( X I I 1 ) ) sin A110 - X I I sin(X11 - 2)O

+ Q I I ( X I+I 1))cos A110 - X I I cos(X11- 2)O


Table 4.5. Singular exponents

correspond to the most singular Mode 1 and Mode 2 displacement components, where

cos ((XI - 1 ) w / 2 )

+

cos((X1 l ) w / 2 ) 1 -- - XI sin((X1 - 1 ) w / 2 ) 1 XI s i n ( ( ~ ~l ) w / 2 )

+

The parameter

K

+

depends only on Poisson's ratio v,

K

= 3 - 4v

K=-

3-v l+v

for plane strain, for plane stress.

Listed in Table 4.5 are values of singular exponents XI and XII for some typical angles. For the convenience of later use. let us denote

158


It is easy to verify following formulas:

Note that when the angle is 360" or 270°, both & ( u Iand ) & ( u Iare I ) unbounded around the comer, whereas when the angle is 240°, 225" or 210°, only the most singular Mode 1 strain components are unbounded around the corner.

EXAMPLE 4.11 We consider a nonlinear Hencky elasticity problem as shown in Figure 4.4, where homogeneous natural boundary conditions are not written explicitly. The part of the boundary represented by thick line segments is fixed. Assume the stored energy function is given by (4.31) and (4.48). Owing to the symmetry of the problem, we consider only a quarter problem, shown in Figure 4.5, where again homogeneous natural boundary conditions are not written explicitly. We are going to use estimates (4.51) and (4.52) to examine how much error we are making if we take the material as a linear one. Let a = 5, b = 4, c = 4, h = 0.5. By using sufficiently accurate finite element analysis, we obtain that for the L-shape singularity (approximately) where a = la,,1 . The strain energy is, approximately,

where E is the Young's modulus. Around the L-shape comer, c(uO) is mainly determined by the first two terms in the expansion (4.53), since strain components for uoare negligible. Hence,


Figure 4.4. Setting of a nonlinear elasticity problem

Figure 4.5. The working problem

we use the first two terms to approxiamtely represent the solution uo of the linear problem for producing estimates for error in the (nonlinear) energy between the solution uo and the solution u of the nonlinear problem, the corresponding estimates from (4.51) andlor (4.52) being denoted by EeSt. In the following, we take

then the linear energy of the linear solution is 0.502041 x ( 2 0 ) ~ .With u = 50, the linear energy is 5.020 x lo2. We compute the error estimates for various /3 and y.See Table 4.6 for the numerical results.


Table 4.6. Example 4.11, error bounds for various parameters

4.3.

LINEARIZATIONS IN HEAT CONDUCTION PROBLEMS

Besides the boundary conditions considered in Section 3.6, nonlinear boundary conditions arise in certain situations, e.g., thermal radiation boundary conditions with heat transfer obeying the fourth-power temperature law, or the natural convection boundary condition with heat transfer proportional to the (514)-power of temperature difference. Provided that the temperature differences are not big the nonlinearities associated with these boundary conditions are avoided by approximating them with linear boundary conditions. We will present a posteriori estimates for the errors caused by such approximations. We will also consider the effect on solutions of solving a heat conduction problem with constant thermal conductivity while the actual thermal conductivity varies slightly within the solid, or may even depend on the temperature.

4.3.1

TEMPERATURE-DEPENDENTTHERMAL CONDUCTIVITY

In certain applications, the thermal property of a material depends on the temperature. Here we consider a case where the thermal conductivity depends on the gradient of the temperature, i.e., we consider the nonlinear differential equation problem


where

f

E L~(R),g E ~ ~ ( 0 ) .

In practice, one usually replaces the nonlinear problem (4.54)-(4.55) by solving a linear problem (linearization) or a sequence of linear problems (see the discussion on the KaEanov method next chapter). We assume the thermal conductivity does not change dramatically with respect to the gradient of the temperature, i.e., k ( z , 1 ~ ~% 1~ o~ ( x) ) , (4.56) and approximate (4.54)-(4.55) by the linear problem

Let us impose some conditions on the coefficient function k for the existence and uniqueness of a weak solution of the problem (4.54)-(4.55), as well as for the a posteriori error estimate on (T - T o ) To this end, we assume k (2,J) is continuous on R x R+ (R+ = [0, cm)), inf{k(x,[) : x E R, J E R+) sup{k(x,J) : x E R,

5 E R+)

> 0, < cm;

(4.59) (4.60)

the function

is strictly convex for a.e, x E R, and there exists a positive-valued function k(x) > 0 such that Vx E R, V E , 7 E IRd, -

Wenote that if k ( x , z ) is differentiable with respect to z , then (4.61) is equivalent to the inequality

for a.e, x E R. Under the above assumptions, one can slightly modify the proof of Theorem 25.J in [I611 to conclude that there is a unique T E H1( 0 ) with T = g on dR such that

Due to the closeness of the functions k and ko by (4.56), from (4.59) and (4.60), it is natural to assume

162


Then applying the Lax-Milgram Lemma, we conclude there is a unique weak solution of the boundary value problem (4.57)-(4.58): To E H ' ( R ) , To = g on dR such that

Let us find a bound for the error ( T - T o ) ,using the solution To of the linear problem (4.57)-(4.58). To do this, we first express the nonlinear problem (4.54)-(4.55) as an equivalent minimization problem. For this purpose, take

V = H ~ ( R )V, * = (H1(C2))*, Q = Q* = ( Av = V v , and where =

{

~~(n))~,

-' fvdx

+cc

ifv=gondR, otherwise.

It is easy to verify that the weak formulation (4.62) is equivalent to the minimization problem

T E V , J ( T , A T ) = inf J ( v , A v ) vEV

(4.63)

For the a posteriori error estimation, we need the conjugate function of J :

I

+cc

2 Jo otherwise,

'"\'"? $ 1

where the constraint set is

and q = -k* ( x ,q * ) is the unique solution of the equation

-q* = k ( x , lq12)q.

163


Now we consider the quantity D(T,T o )defined in (2.12). An upper bound of the difference, by Theorem 2.40, is

J(To,ATo)

+ J*(A*q*,-q*) = /R [q*. (Vg + k * ( x q, * ) )- f

(To - g )

for any q* E Q:. For a lower bound, we first note that from the definition of G,

J k(x,

W P ) , q) =

R

I P ( X ) I ~PW )

. q ( x )dx.

Recall the assumption (4.61). Use the formula (2.20),

D(T,To) =

1'

[ k ( x ,IVT

R 0

+ t V ( T o- T ) J (VT ~ ) + t V ( T O- T ) )

Thus, we have the following error estimate:

Snle(x)IWTo

i.e., where the membrane touches the obstacle is an unknown.

184

4.7.2


LINEARIZATION, A POSTERIORI ERROR ESTIMATE

It is advantageous to make simplifications in modeling a real problem wherever possible, and it is highly desirable to know quantitatively how much error is introduced through the simplifications, once the simplified mathematical problem is solved. The goal of the section is to provide such an a posteriori error estimate for linearization. Let us drop the constraint "v 2 $ in R" for an admissible function, and minimize the energy function E ( v )over all v E H 1( 0 )satisfying v = g l on r l . In other words, we introduce the simplified problem of finding uo E Hhl,gl( 0 ) such that

The corresponding variational equation problem is to find uo E that

L V u o . V v d x = L f v d e + L g2vds

~h~ ,gl ( 0 )such

VVEH:~(R).

2

This is the weak form of the linear elliptic boundary value problem

Our question now is whether Go = max{uo,$1 can be accepted as an approximate solution of the obstacle problem (4.101). The reason we choose Go instead of uo is that Go E K, whereas uo does not have this property. Let us derive an a posteriori error estimate for ( u - G o ) . For this purpose, we introduce the spaces:

the linear operator A E L (V,Q ):

and the energy functional

J ( v ,Av) = F ( v )

+ G(Av),


where F(v) =

{ ;if

v E K, otherwise,

vdx9

Obviously, E ( v ) = J ( v , Av), v E K. Consider the quantity D ( u , v) given by the formula (2.21). We notice that for P, q E Q3

Then by the formula (2.2I), we have

Now we turn to an upper bound for D ( u , v). We calculate the conjugate functionals. F*(A*q*)= sup [(Av,q*) - F ( v ) ] vEV

= sup [/,(q?

n u

WE K

={& Thus,

+f

.)dx

+

l2

n w s ]

1

q ; I2dx

if qz = g2 on T2, otherwise.

186


In the following, we make the particular choice

For any w E H : ~

(R), from the boundary value problem for UO,we obtain

Thus,

and we can modify the formula for J*(A*q*,- q * ) accordingly. Therefore, applying Theorem 2.40, we get the following upper bound for D ( u ,Go):

From this inequality and (4.102), we have thus proved the following general (R), error estimate: for any w E H;,,,,

In particular, if we take w = uo, then we have the a posteriori error estimate:


187

And if we take w = fro, then

We will see later through a detailed analysis for a one-dimensional problem, that the a posteriori error estimates (4.104) and (4.105) provide efficient error bounds.

4.7.3

DISCRETE A POSTERIORI ERROR ESTIMATE

For simplicity, we will describe the results for triangular linear elements for an obstacle problem on a polygonal domain a. The generalization of the results for higher order elements, rectangular elements and higher dimensional problems on general domains is straightforward. In particular, our discrete a posteriori error estimates hold independent of concrete structures of finite element spaces. Thus, we triangulate into finite number triangles satisfying the standard condition that two distinct triangles from the triangulation either are disjoint, or have a common vertex or a common edge (cf. Section 1.8). We further assume and are vertices of the triangulation. Let h that the intersection points of be the maximum length of the triangle edges. Assume the pointwise values of gl and '$ exist. We let Nh denote the set of all the nodes (the triangle vertices in 2), Nh,rlthe set of those nodes lying on rl, and Nh,o= Nh\Nhlrlthe set of free nodes. We use the finite element set

Vh = { v h E ~ ' ( 0 : vh) is linear on each triangle, v h ( a ) = g l ( a ) v a E Nh,rl)

to approximate

HI^ (a), and ,gl

to approximate K. Then the finite element solution of the obstacle problem (4.100) is uh E Kh such that

E(uh)= inf E(vh) ~h€Kh

which is equivalent to the constrained quadratic programming problem (cf. [ 6 5 ] )

where A = ( a i j ) E IRNxN is symmetric positive definite, v = ( v l ,. . . , v ~ ) ~ , N is the number of nodes in Nh,0,and C is the constraint set


188

0 and 0 < k < 1 can be computed from the problem setting or estimated from several iterates. Therefore, with the knowledge on the size of the initial error, it is easy to find the number of iterations needed for achieving required accuracy. As in the case for continuous problems, we introduce the finite element solution of a linearized problem. We drop the constraint in the finite element set K h , and use Vh to define a finite element solution of the linearized problem, - u ~ , hE Vh such that E(uO,h)= inf E ( v h ) . ~hcvh

In practice, it is easier to solve (4.108) than (4.106). Especially when the given data are of certain special type, fast solution method exists for solving (4.108). E Kh After we get the solution uo,h, we define an approximation solution by f i o , h ( a ) = max{uo,h(a), '$(a)) v a E N h , o . (4.109) We can then derive the following discrete versions of a posteriori error estimates (4.104) and (4.105):

Now, a modified over-relaxation method with projection for solving the finite element system (4.108) consists of three steps. In the first step, we solve the linear problem (4.108), and define an approximation through (4.109). In the second step, we apply one of the above two a posteriori estimates to see how big is the error //V(fio,h- uh) 1 1 L 2 ( n ) . If the error bound is smaller than the error tolerance, we accept as the finite element solution of the obstacle problem, uh, and stop computation. Otherwise, we go to the next step. In the third step, we take as the initial guess, and apply the over-relaxation method with projection to solve (4.106). Here, we also use the computed bound on IIV(fio,h- uh) 11 L 2 ( n ) to decide how many iterations we need to get an accurate solution uh.

190

4.7.4


A ONE-DIMENSIONAL EXAMPLE

Let us consider a one-dimensional obstacle problem. We take R = ( 0 ,I ) ,

rl = F = d R , r2= 0,$ = 0, and the admissible set

K = { v E H ; ( R ) : v 2 Oin ( 0 , l ) ) . Given f E ~ ~ I( ) ,0the, one-dimensional obstacle problem is to find u E K such that E ( u ) = inf E ( v ) , (4.110) vEK

where the energy function is

Equivalently, u E K is the unique solution of the variational inequality:

To make specific computations we use the following form of the force function 2 (4.111 ) f(x)=--J-X, 3 where the parameter J E [O, 2/31, We first derive an analytic expression for the solution u of the obstacle problem (4.110)when f is given by (4.111). Formally, the solution u of the obstacle problem (4.110) solves the following boundary value problem:

- u l ' > f , ~ 2 0 ,( - u t l - f ) u = O i n n , u = O o n r , u = O and u l = O on F*, where r*is the "interface" between the sets { x E R : u ( x ) = 0 ) and { x E R : u ( x ) > 0 ) . We can obtain the following formula for the (unique) solution of the obstacle problem:

As for the solution uo of the corresponding linear problem for (4.1l o ) , we have 1 u o ( x )= - x 3 (4.112) 6 3


Hence, our linear approximation Go = max{uo,0) to the solution u is

From the error estimate (4.104), we have the following a posteriori error estimate: 1

1

1

( ~ ~ - u I )5 ~ d x [Gf -uf - 2 0

(:

- J - L )

(Go - u o ) ] dx. (4.113)

We have a similar error estimate from (4.105). We note that the size of the parameter J determines the closeness of the linear problem to the obstacle problem. In particular, when J = 0, the two problems coincide, and the linearization causes no error. When f > 0 is close to 0, we expect the solution of the linear problem can be used as an approximation of the obstacle problem. The estimate (4.113) offers an upper bound for the error (Go - u), employing the solution uo of the linear problem only. We will see how accurate is our a posteriori error estimate (4.113) as compared to the true error. Since we have analytic expressions for u, Go and uo, we can compute both sides of the estimate (4.113) exactly. Thus, for the model one-dimensional obstacle problem (4.1 10) [with f given by (4.11I)], we can compare our a posteriori error bound with the exact error. Now let us compute the true error

The error upper bound from (4.113) is

We observe that as J -+ 0+, the approximation error 11 Gb - ulII L 2 ( n ) -+ 0, and the ratio between our error bound for / / f i b - ~ ' l l ~ 2 (and ~ , the exact error 11 Gb - u111 L2 cn)tends to

Thus, our a posteriori error estimate provides a quite accurate quantity for the exact error 11 Gb - ull/L2(n), when f is close to 0.


Table 4.8. Efficiency of error bounds on a one-dimensional obstacle problem

Table 4.8 shows some numerical results. Numerical results from the error estimate (4.105) are similar, and are omitted here. Although the above example is for the effectiveness of our a posteriori error estimates for continuous problems, it also indicates the effectiveness of the a posteriori error estimates for finite element solutions, for the discrete solutions will be sufficiently close to the continuous solutions when the finite element mesh is sufficiently refined.

Chapter 5

A POSTERIORI ERROR ANALYSIS FOR SOME NUMERICAL PROCEDURES

In this chapter, we apply the duality technique to derive a posteriori error estimates for some numerical procedures, including the regularization method and the KaEanov iteration method, in solving nonlinear boundary value problems. The regularization method is usually employed in numerical treatment of problems involving non-smooth terms. One family of such problems is the variational inequalities of the second kind. The KaEanov iteration method provides a sequence of linear problems to approximate a nonlinear problem, and can be quite efficient when the nonlinearity is not strong. For practical implementation of the regularization method or the KaEanov method, it is highly desirable to have some a posteriori error estimate so that once a regularization iterate or KaEanov iterate is computed, we can compute a (presumably efficient) error bound for the approximate solution with ease. Such an error estimate can provide us some information about the reliability of numerical solutions, and can also be used as a convenient stopping criterion for iterations. Our presentation of the a posteriori error analysis here is focused on the continuous problems, and can be easily adapted to treat the discretized problems, as is demonstrated in Section 5.4.

5.1.

A POSTERIORI ERROR ANALYSIS FOR REGULARIZATION METHODS

The purpose of the section is to give an a posteriori error analysis of regularization methods for solving non-differentiable minimization problems. The model problem to be discussed is a generalized version of an obstacle problem considered in [132]. Let R C IRd be a Lipschitz domain. In the study of obstacle problems for applications, d = 2. However, all the arguments 1 be ~ below are valid for any dimension d. Let f E L~( 0 )and g E ~ ' (OR)

194


given non-negative functions. Denote the energy functional

and the admissible set

K = { v E H ~ ( R: )u 2 0a.e. i n n ) , where

H$)

=

{v E ~

l ( n : v)= g

on an).

Then the obstacle problem is

E ( u ) = inf E ( v ) .

u E K,

WEK

(5.2)

The problem is equivalent to an elliptic variational inequality of the first kind:

Such an obstacle problem describes the vertical displacement of a thin elastic membrane that has a prescribed height g on its boundary, lies above a planar obstacle, and is subject to the action of a vertical force with a scaled density - f . By the non-negative assumption on f , the vertical force acts downward. Existence of a unique solution of the problem follows from the standard result on the unique solvability of variational inequalities of the first kind (Theorem 1.24, page 29). To develop the regularization method, we write the obstacle problem in another form.

T H E O R E5.1 M Denote

b ( v )=

1 (i R

j0v12

+f

v I ) dx.

Then the problem (5.2) is equivalent to the problem ofjnding u E H: ( 0 )such that E ( U ) = i n f { ~ ( v: )v E H ~ ( R ) } . (5.4)

Proof. We first show that the problem (5.4) has a unique solution. Since g E H112(dfl), it is the trace of an H 1 ( R )function, which is denoted by the same symbol g. Let us introduce a change of variable

vo = v

-g

for v E ~ ' ( 0 ) .

A Posteriori Error Analysis for Some Numerical Procedures

Then a solution of the problem (5.4) is

where uo E

HA

(R) minimizes the functional

HA

(0).This problem is equivalent to an elliptic among all the functions vo E variational inequality of the second kind: Find uo E (0)such that

HA

where

It is easy to verify that a(., -) is a continuous, H: (a)-elliptic bilinear form, j ( . ) : H; (R) + R is proper, convex and continuous, and E is a continuous linear form on ( a ) . So the variational inequality (5.6), and thus the problem (5.4), has a unique solution (Theorem 1.24 in Section 1.7). To prove the equivalence of the problems (5.2) and (5.4), we use the following result from Lemma 1.19 (page 22): If v E H1(R), then 1 v 1 E H' (0), and

HA

For the solution u of the problem (5.4), we have lul E H 1 ( R ) , lul = g on dR, and by (5.7), E(lul) L ~ ( 4 . By the uniqueness of a solution of the problem (5.4), we thus have u = lul 2 0 in R. Hence, the solution of (5.4) is also the unique solution of the problem (5.2). rn Recalling the relation ( 5 3 , we see that the solution u E H ~ ( R )of the problem (5.4) satisfies the variational inequality

f ( I V- I U ) dx 2 o 'dv E H;(R).

(5.8)

196


A major difficulty in solving the problem (5.8) numerically is the treatment of the non-differentiable term Ja f dx. In practice, there are several approaches to circumvent the difficulty. One approach is to introduce a Lagrange multiplier for the non-differentiable term, and the problem (5.8) (and its discretization) is solved by an iterative procedure, for detail, see, e.g., [65]. In this section, we will give a detailed analysis of another approach, namely, the regularization method. The idea of the regularization method is to approximate the non-differentiable term by a sequence of differentiable ones. The regularization method has been widely used in applications (cf. [65], [66], [94], [131]). An approximating differentiable sequence in the regularization method depends on a small parameter E > 0. The convergence is obtained when E goes to 0. However, as E + 0+, the conditioning of a regularized problem deteriorates. So, there is a tradeoff in the selection of the regularization parameter. Theoretically, to get more accurate approximations, we need to use smaller E . On the other hand, if E is too small, the numerical solution of the regularized problem cannot be computed accurately. Thus, it is highly desirable to have a posteriori error estimates which can give us computable error bounds once we have solutions of regularized problems. We can use the a posteriori error estimates in devising a stopping criterion in actual computations: If the estimated error is within the given error tolerance, we accept the solution of the regularized problem as the true solution; and if the estimated error bound is large, then we need to use a smaller value as the regularization parameter E . An adaptive algorithm can be developed based on the a posteriori error estimate.

VI

5.1.1

THE REGULARIZATION METHOD

In a regularization method, we approximate the non-differentiable term j (vo) by a sequence of differentiable ones,

The regularized problem is to find uo,,E H; (R)such that

or u, E H: (R)such that

+

The relation between the solutions of the two problems is u, = uo,, g. We expect under certain conditions, uo,,-+ uo (equivalently, u, + u)as E + 0. See the discussion below.


197

For a given non-differentiable term, there are many choices for a sequence of differentiable approximations. Let us list five natural choices of a regularizing sequence for the obstacle problem (the first two choices are taken from [94]).

Choice 1. j, (vo)=

Choice 2. j, (vo)=

Choice 3. j, (vo)=

Choice 5. j, (vo)=

f

I

$2 (vo+ g) dx with

f $:(vo

+ g) dx with

+

f

(vo g) dx with

S, 4:

(vo g) dx with

/a

f

=

+

{ (-

1 t2

2

E +&)

i i t

< r,

Following [66], we present a general convergence result on the regularization method.

L E M M A5.2 Let V be a Hilbert space, a : V x V -+ R bilineal; continuous and V-elliptic, j : V + R propel; convex and I.s.c., l : V -+ R linear continuous. For varepsilon > 0, assume j, : V + R ispropel; convex and 1.s.c. Assumefirther that for some constant cl, allowed to be negative, jE(v) > c1 'dv E V,


198

For relations between j

( a )

and j,

(e),

we assume

v, -+ v weakly in V

j(v) I lim inf j,(v,). &+O

(5.12)

Let u , u, E V be the solutions of the variational inequalities

and

respectively. Then, u,

-+

u in V, as E -+ 0.

Proof. In the variational inequality (5.14), we take v = 0 to get

>

By the assumptions, j,(u,) cl. Since j,(O) -+ j(O), we see that {j,(O)) is bounded for E small. Therefore, by the V-ellipticity of a(.,-), we conclude from the above inequality that {IIu,/lv) is bounded for E small. Then we have a subsequence, still denoted as {u,), and u E V, such that

Let us take the limit E -+ O+ in (5.14). By assumptions, V v E V,

Finally, from

we obtain

liminfa(u,,u,) e+O

> a(u,u).

Using all these relations, we see that the limit u satisfies the variational inequal1 ity (5.13). For the obstacle problem and its regularizations presented above, we can verify that all the conditions of Lemma 5.2 are satisfied. In particular, let us


199

show the conditions (5.11) and (5.12) and do these first for the choices 2 and 3. We notice that by the compact embedding theorem (Theorem 1.13, page 14), the space V is compactly embedded in Lp(R) for some p = p(d) > 2; for d = 2, p > 2 can be arbitrary, and for d 2 3, 2 < p < 2d/(d - 2). Let EO = (p - 2 ) / 2 > 0 and vo E Hi( 0 ) .Then for E < EO,we have

By the Holder inequality, the right hand side function is integrable:

Also, for both choices 2 and 3,

Applying the Lebesgue dominated convergence theorem (see, e.g., [18, page 2461 or most other textbooks on real analysis), we obtain the limiting relation v in V implies the existence of a (5.11). To show (5.12), we note that v, subsequence, still denoted by {v,), such that

-

v,

-+vo

in L P ( R )and a.e. in R.

We can then similarly apply the Lebesgue dominated convergence theorem to obtain (5.12), where the inequality is actually an equality:

j ( v o ) = lim j&(v,).

&to

The verification of (5.11) and (5.12) for the choices 1, 4 and 5 can be done similarly, and is actually simpler. It is not difficult to show that for any t E&!FI

Then for the choices 1 and 5,

and for the choice 4,


200

We have seen that for some choices of the regularization function, there is an inequality of the form

for some constant co. Under this additional assumption, we can actually have an error bound on (u- u,)for solutions of the variational inequality (5.13) and its regularization (5.14). We take v = u,in (5.13) and v = u in (5.14) to obtain

Add these two inequalities,

.) (denoting a the ellipticity constant) and the Using the V-ellipticity of a(., assumption (5.15), we get

5.1.2

A POSTERIORI ERROR ESTIMATES

As far as practical computation is concerned, a convergence result and an a priori error estimate are not enough for a complete numerical analysis with the regularization method. A posteriori error estimates are more desirable that will provide a quantitative error bound once a solution of the regularized problem is computed. In this regard, when the condition (5.15) holds, we can use (5.16) for an error bound. In this subsection, we will use the duality theory to derive some other a posteriori error estimates. We will first give an analysis in the framework of a general regularization method, and then discuss the applications to the regularization method with various choices of the regularization sequence. For the obstacle problem (5.4) considered in this section, we use the function spaces

the operator

Av and the functional

=

(VV~V)~

J(v,Av) = F(v)+ G(Av),


where F(v) =

0 oo

ifv = g o n d R , otherwise,

d

As before, we use the notation q = (q,, q2) for q E Q, with q, E ( ~ ~ ( 6 2 ) ) and 92 E L~( 0 ) . A similar notation is used for q * E Q*. With the above notations, the obstacle problem (5.4) can be rewritten in the form of (2.5). To apply Theorem 2.40 for a bound on (u, - u), we first compute the conjugate of the functional J. We have F * ( A * q * ) = sup / ( q ; - V u + v q j ) d z VEH;(R) R

and

-

lq212

dx if 1qz1

< f a.e. in 0,

otherwise. Hence, denoting the constraint set QE = {q* E Q* : L ( q ; V V a

+ qjv) dx = 0 Vv E H ~ ( R ) ,

we have

\

otherwise.

Since G(q) is not Giiteaux differentiable, we consider the energy difference

202


Using (5.8) with v = u,,we find that

On the other hand, applying Theorem 2.40, we have

for any q* E Qz. Hence, we have the general inequality for a posteriori error analysis:

Let us then turn to the regularized problem (5.10). Since 4,is differentiable, the variational inequality (5.10) is equivalent to the equality

Thus u,is the weak solution of the elliptic boundary value problem

+

-nu, f (4,)'(u,) = 0 in R , u,= g o n d R . From (5.19), we observe that if the regularizing function 4 satisfies the inequality ' I 5 1 vt E 4 (5.20) then a natural selection of an auxiliary field q* in the basic inequality (5.18) is

From (5.18), the following a posteriori error estimate is obtained:

Taking v = u,- g E HA (R) in (5.19), we find that

Sn

[V% . V(%

- g)

+ f (4E)'b.)

(U&- g)] dx = 0.

Therefore, we can write the a posteriori error estimate in the form of


203

The regularizing functions in the choices 1, 4, and 5 satisfy the inequality (5.20). Hence, we have the following a posteriori error estimates. For the choice 1, we have

Thus the a posteriori error estimate is

For the choice 4, we have

Thus the a posteriori error estimate is

For the choice 5, we have

So we have the same form of a posteriori error estimate as that given in (5.23). For the choices 2 and 3, the regularizing functions do not satisfy the inequality (5.20). It is still possible to construct an admissible field q* from u,to produce a good error bound, but the procedure will be more involved.

REMARK5.3 The material of this section is based on the reference [87]. An earlier work on a posteriori error analysis of regularization methods for a holonomic elastic-plastic problem is given in [69]. Similar a posteriori error estimates have been derived for regularization methods in the more complicated situation of elastoplasticity problems. See [75, 77, 78, 79, 801.

5.2.

K A ~ A N O VMETHOD FOR NONLINEAR PROBLEMS

The KaEanov method is an iteration method for solving nonlinear problems, via linearization. An early reference on the method is [91], where the method is

204


applied to compute a stationary magnetic field in nonlinear media. Convergence of the method is proved in the context of the particular application there, though the technique of the proof is rather general. A numerical example in [91] shows that the method is quite effective for certain problems. See also [58]. Later, the method is applied to a nonlinear elasticity problem in [I 111. For applications of the method in solving variational inequalities for transonic flows in gas dynamics, see [53, 641. A general convergence result of the method is presented in [I121 (see also [161]) for solving a nonlinear variational inequality of the first kind. In [74], convergence of the method is proved for solving an even more general nonlinear variational inequality of the mixed kind (i.e., it is an inequality both because the problem is posed over a non-empty closed convex subset and because the problem contains non-differentiable terms), and the result is used in solving a nonlinear variational inequality of the second kind arising in elastoplasticity. The main purpose of the section is to introduce the KaEanov method and present a rather general convergence result. In the following sections of the chapter, we will derive a posteriori error estimates for KaEanov iterates in the context of concrete application problems. Whenever we discuss the KaEanov method, no summation is implied over the repeated index k for the KaEanov iterates. As an introductory example, let us consider the problem

where R c &idis a Lipschitz domain with its boundary dR = rl U E, Fl relatively closed, F2 relatively open, and rl r7 F2 = 0.A problem of the form (5.25) was considered in Subsection 4.1.1. The boundary value problem (5.25) is a stationary conservation law type problem. We will impose conditions on the data later for the unique solvability of the problem. The KaEanov method for solving (5.25) is the following. Given an initial guess uo with uo = g on F l , we define a sequence of iterates {uk)k>l - by the boundary value problems:

for k = 1 , 2 , . . . . Notice that in (5.26), we solve a sequence of linear problems instead of the nonlinear problem (5.25).


205

We now provide a rather general convergence result on the KaEanov method. The presentation here follows [39]. Let V be a Hilbert space, K c V a nonempty, closed, convex subset. Let A : K + R be Giteaux differentiable, j : K + R be non-negative, nondifferentiable, convex and continuous, and t E V* a continuous linear functional on V. Define the energy functional

Then we consider the minimization problem:

U E K , E ( u ) = inf E ( v ) . VEK

(5.27)

Since A is Giteaux differentiable and j is convex, a solution of (5.27) satisfies the variational inequality

When A(v) is not quadratic in v, the first term ( A 1 ( u v) ,) in (5.28) is non-linear in u. Assume there is a functional B : K x V x V + R such that

and for fixed u E K , B ( u ;v , w ) is bilinear with respect to v and w. The problem (5.28) can be rewritten as

Suppose { K i ) z ois a sequence of nonempty, closed and convex subsets of K with properties that Ki C Ki+l, i = 0,1,. . ., and U E oKi is dense in K . Then the KaEanov method for (5.30) is defined as follows: Choose uo E KOas an initial guess. For k = 1 , 2 , . . ., we find uk E Kk by solving the problem

In the conventional KaEanov method, KO= K1 = . . - = K . Here we have a more general setting by allowing the use of a sequence of nested subsets.

T H E O R E5.4 M We keep the above assumptions on the given data V , K , {Kk), A, j, !and B. (a) Assume for each u E K , the bilinearform ( v ,w ) H B ( u ;v , w ) is symmetric from V x V to R.Also, assume there are constants S1 > 0 and So > 0 such that (5.32) lB(u;v , w)l I 61 llull llwll V U E K , V v ,w E v,

206


and

Then the problem (5.31) has a unique solution uk E K k , which is also the unique solution of the minimization problem, 1 - B ( u k W lv; , v ) 2

+ j(v) - t ( v )-+

inf,

v E Kk.

(5.34)

(b) Further assume A' : K -+ V * is continuous and strongly monotone, i.e., there exists a constant po > 0 such that

Also, assume the following key inequality:

A(v) - A(u) 5

1 ( B ( u ;v , v ) - B ( u ;u, u ) ) 'v'u,v E K. 2

-

(5.36)

Then (5.27)has a unique solution u E K, which is also the unique solution of the variational inequality problem (5.28). The KaCanov method defined in (5.31 ) converges, i.e.,

Proof. The equivalence between the problems (5.31) and (5.34), or between the problems (5.27) and (5.28), can be established in a standard way; cf. Examples 1.26 and 1.27. From the stated assumptions, we can verify that for each k 2 1, the functional

is strictly convex, 1.s.c.and weakly coercive on Kk (i.e., Ek ( v ) -+ oo as llvli -+ oo, v E K k ) . Thus, by Theorem 2.37, the minimization problem (5.34) has a unique solution uk E K k . Similarly, the problem (5.27) can be shown to have a unique solution. Now we prove the convergence of the KaEanov method defined by (5.31). We consider the energy sequence { E ( U ~-) ) Using ~ > ~ (5.36) . and (5.31), we


have

Therefore, the sequence { E ( u k ) )is decreasing. Since { E ( u k ) )is bounded below by the minimum value of E ( v ) over K , we have lim [ E ( U ~ - ~E )( u k ) ]= 0.

k+co

Then, from (5.37) we have lim lluk - u ~ - ~= (0./

k+w

Using (5.35) and (5.28), we get, for any v E Kk+l,

From (5.31),

Hence,

u ) ) ~ < I I +inf Iz(v), ~€Kk+l

(5.39)


208 where

and 12(v) = B ( u k ;uk+l, v

- U)

+ j ( v ) - j ( u ) - t(v - u).

Il = 0. By (5.38) and the fact that {uk) is bounded in V, we see that limk,, Using (5.32) and the continuity of t , we know that there exists a constant c such that inf I12(v)l 5 c inf (llv - uII lj(v) - j(u)I). vEKk+l

Since limk,,

uEVk+1

+

Kk = UkKkis dense in K , we see that lim I z ( v ) = 0.

k+,

Therefore, limk,, Iluk - ull = 0 by (5.39). Theorem 5.4 extends various convergence results found in the literature as special cases. For example, if we let j ( . ) = 0 and take Ki = K for each i, then we recover the convergence result presented in [161]. Theorem 5.4 also generalizes the convergence result proved in [74]. We introduce the sequence {Ki) for more flexibility in applications of the method. We may, for instance, progressively enrich the finite element subsets while we perform the KaEanov iteration. More precisely, we can combine the finite element approximation and the KaEanov iteration in solving the problem (5.30). Let {Ki) be a sequence of finite element subsets such that Ki c Ki+l and Ui Ki is dense in K . Then (5.31) offers a numerical algorithm where for each k, we solve a finite dimensional linear variational inequality for uk, and as k + oo, we have the convergence uk + U. Although this coupling of the finite element method and the KaEanov method is not elaborated further in this work, it is worth exploring the possible efficiency gain offered by such a coupling. In the following, we focus on the use of the standard KaEanov method, i.e., where KO= Kl = - - . = K . For (5.28), the method is defined as follows. Let uo be an initial guess chosen from K. For k = 1 , 2 , . . . , we find uk E K such that

As a corollary of Theorem 5.4, we have the next result concerning the convergence of the standard KaEanov method.

T H E O R E5.5 M We keep the stated assumptions on the given data V, K , A, j,

t and B.

209


(a) Assume for each u E K , the bilinearform ( v ,w ) t, B ( u ;v , w ) is symmetric from V x V to R . Assume there are constants S1 > 0 and So > 0, such that

and

>S~IIV-W~~~

B(u;v-w,v-w)

Vu,u,wEK.

(5.42)

Then the problem (5.40)has a unique solution uk E K , which is also the unique solution of the minimization problem 1 2

- B ( u k P 1v; , v )

(b)Further assume A' : K for a constant po > 0,

+ j ( v ) - t ( v ) + inf,

v E K.

+ V * is continuous and strongly monotone, i.e.,

Also assume the following key inequality: 1

A ( v ) - A ( u ) 5 I[ B ( u ;v , v ) - B ( u ;u , u ) ] V u ,v E K . 2

(5.44)

Then (5.27) has a unique solution u E K , which is also the unique solution of the variational inequality problem (5.28). The KaEanov method deJined in (5.40) converges, i.e.,

uk

5.3.

+u

in V , a s k

+ co.

K A ~ A N O VMETHOD FOR A STATIONARY CONSERVATION LAW

In this section, we consider the KaEanov meoth (5.26) for the stationary conservation law (5.25), derive an a posteriori error estimate for the KaEanov iterates, and present numerical examples to show the effectivity of the KaEanov method and the error estimate. Let us introduce the set

where and the functionals

v =H~(R),

210


where

We have with

P

B ( u ;v , w ) =

/a

a(lVu12)vv V w dx.

Then, the problems (5.25)for u E K and (5.26)for uk E K can be rewritten as

A ( u ) - t ( u ) = inf { A ( v )- t ( v ) }, UE

1 2

-B

(5.45)

K

( U ~ -uk, ~ ;u k ) - t ( u k )= inf

B ( u k - ~v;, v ) - t ( v )

The variational equation for uk E K is

We introduce the following assumptions.

( A l ) R c IRd bounded and Lipschitz continuous, H ' ( R ) , and h E H 1 I 2 ( r 2 ) * . ( A 2 ) The function a : R+ and Po, such that

rl #

a), f E H 1 ( R ) * ,g E

+ R+ is C1, and there are positive constants a l , a2

Here in ( A l ) , we assume that the given Dirichlet datum g is the trace of a gE ( R ) on rl. We note that ( A 2 )is a standard assumption in applications such as nonlinear magnetostatic field problem ( [ g l j )or elastoplasticity problem (11 121)The following result is derived from Theorem 5.5 (see also [I611).

T H E O R E5.6 M Assume ( A l ) and ( A 2 ) Then, (5.45) has a unique solution u.Let uo E K be given. Then for k = 1,2, . . . , (5.46) has a unique solution uk, and uk + u in H 1 ( R ) ,a s k + m. We now turn to an a posteriori error analysis for the KaEanov iterates. We note that a different a posteriori error estimate is given in [155],also via duality

21 1


theory, but in a different manner. That estimator is seemingly more involved to compute as it requires the computation of another linear problem roughly of the same size as the one required to produce the most recent term uk in the KaEanov sequence. There, no numerical results are given. Let

Q=( ~ Av = Vv,

~(n))~"~,

and identify Q* with Q. Define a functional from V x Q to E: J(v, 9 ) = F ( v ) + G ( 4 , where, F(v) =

{

- f v d x - ~ h v d s ifvtK, otherwise,

+m


Then, from the definition of a conjugate function, we can find Q* . Vg

+ f g + Iq*l b(Iq*l)

J*(A*q*,-q*)=

ghds otherwise,

+m

where t = b(lq*I) is the unique solution of the equation / q * /= a ( t 2 )t. From the definition of G ( q ) , we have, for p, q E Q, P

ifq*EQz,

212


~ using linear elements on a uniform mesh with 21 nodes on the domain [ O , l ] .

+

EXAMPLE 5.7 Let us consider the case when a ( s ) = 1 + X / ( 1 s ) , 0 5 X < 8. The assumptions ( A 1 )and ( A 2 )are satisfied, with Po = 1 - X/8. In this example, we let f ( x ) = 1, qo = ql = 0 , and uo( x ) = 0. Tables 5.1 and 5.2 show the convergence of the KaEanov method and the effectivity of our a posteriori error estimates for the KaEanov iterates. In these tables, I% is , ~u ; l L l ( o , l ) , ~ h , kis the a posteriori error the iteration number, eh)k = ~ u L estimate for eh,k from the discrete analog of (5.48), and = ~ ~ , ~is / e ~ the effectivity index. The corresponding values for the KaEanov method on the continuous level are denoted by ek, EI, and Bk. The first columns of the tables are iteration numbers. We observe that the KaEanov method converges fast both for solving the continuous problem and for computing the finite element solutions. For the case when X = 1 (Table 5. I), the effectivity indices lie between 1.4 and 1.5, showing the efficiency of our a posteriori error estimates. When X = 6 (Table 5.2), the convergence of the KaEanov method is still fast, while the effectivity indices are not close to 1. This is due to the fact that in our a posteriori error estimate (5.48), the left sides contain a factor of Po. Our a posteriori error estimate always bounds the true error, and hence, we expect the error estimate will become less efficient when certain parameter in a problem tends to a critical value ( A + 8- in the current example).


Table 5.1. Example 5.7, numerical results with a ( s ) = (2

+ s ) / ( l + s ) , Po = 718

Table 5.2. Example 5.7, numerical results with a ( s ) = (7

+ s ) / ( l + s ) , Po = 114

+

+

We have similar numerical results when a ( s ) = 1 X / ( 1 s2), 0 5 X < 1619. In this case, Po = 1 - 9x116. Table 5.3 provide numerical results for the problem with X = 1. H

EXAMPLE 5.8 In this example, we apply the KaEanov method to solve the problem -

[ a ( l ~ ' 1 ~ ) ~ ' ] ' =f o0 r z E [ o , l ] , u(0) = 0,

u(1) = 1.


Table 5.3. Example 5.7, numerical results with a ( s ) = ( 2

+ s 2 ) / ( 1 + s 2 ) , Po = 7/16

This problem has an analytic solution u(x) = x. We choose a family of a ( s ) by following: Let 1 3 3 a ( t ) = -t - -t, 2 2 and positive parameters S and K , for x < 1 - S, forx~[l-b,l+S], for x > 1 S.

+

Then for s 2 - 1 E [-S, S], a ( s )

For s 2 - 1

> - 1, al(s) > -312

and s 2 5 1

+ h so that

> S we have

As n + 0, a tends to 1 uniformly and we might expect = &k/ek to tend to 1. Let yk = ek/ek-l, k = 1 , 2 , . . . , be the error reduction factors. Again, we observe the fast convergence of the KaEanov method, and the efficiency of the a posteriori error estimates. Tables 5.4-5.8 show us results for S = 712 and different K . Note that theoretically, our a posteriori error estimates always produce upper bounds on the exact error. Some of the effectivity index values in the tables are slightly smaller than 1, this reflects the influence of numerical errors resulting from using quadratures in computing both the error estimates and the true error. As expected, for n = 0 we get the solution in one iteration.


Table 5.4. Example 5.8, numerical results with r; = 0.20488

Table 5.5. Example 5.8, numerical results with r; = lo-'

Table 5.6. Example 5.8, numerical results with r; = lo-'

Table 5.7. Example 5.8, numerical results with r; =

217

218


Table 5.8. Example 5.8, numerical results with K =

Table 5.9. Example 5.9, convergence of KaEanov iterates

EXAMPLE 5.9 As a further numerical example, we use the KaEanov method to solve the problem -

[ ~ ( ( u ' ~ )= u '1]0' ~ 0 ~ ( 1 O xfor ) x E [O, 11, u ( 0 ) = 0 , u ( 1 ) = 1.

(5.52) (5.53)

Here, a ( s ) = 712 - arctan(s)/lO. We can verify that the assumptions ( A 1 ) and ( A q )are satisfied, and d = 712 - ( 1 ~ / 4 ) / 1 0For . the initial iterate, we take ex - 1

+

4 4 = p-1.

Figure 5.1 and Table 5.9 show the convergence of the derivative u; for the problem (5.52)-(5.53). In Table 5.9, k is the iteration number, ek = 11u$ u'llL2(0,1),~k is the error bound for ek from (5.48), Ok = Ek/ek is the effectivity index, and yk = e k / e k - l is the error reduction factor. H


Figure 5.1. Example 5.9, convergence of KaEanov iterates

EXAMPLE 5.10 The last example is also interesting. Consider the problem -

[ a ( l ~ ' ~ ) u '= ] ' O for x E [01Ill u(0) = 0, u(1) = 1.

+

Here, a ( s ) = arctan(s) 7r/2. We take u o ( z ) = x2. It is easy to see the condition a l ( s ) 5 0 in (Az) is not satisfied. The other conditions in (A2) and (A1) hold, and we can take d = 71.12. Despite the fact that a does not satisfy conditions (A:,), we still have the convergence of the KaEanov iteration method, and the a posteriori error estimate (5.48) is efficient (see the numerical results reported in Table 5.10 and Figure 5.2).

5.4.

K A ~ A N O VMETHOD FOR A QUASI-NEWTONIAN FLOW PROBLEM

In this section, we extend some of the earlier discussions to cover nonlinear boundary value problems with constraints. We will take as a sample problem that of a quasi-Newtonian flow where the constraint is the incompressibility of the fluid. The discussion here, however, does not depend on the particular form of the problem we solve and can be extended in a straightforward way for other nonlinear problems with constraints. We now describe the problem of a quasi-Newtonian flow obeying the Carreau law. Let R c IEd be a Lipschitz domain. The boundary value problem


Figure 5.2. Example 5.10, convergence of KaEanov iterates

Table 5.10. Example 5.10, convergence of KaEanov iterates

governing a quasi-Newtonian flow is ([14, 151)

S,

p d x = 0.

22 1


) the ) ~ velocity field and ~ ( u: R ) Here u ( x )= (ul( x ) ,. . . , u ~ ( x is the rate of linearized deformation tensor with entries

-+ sdis

The unknown variablep is interpreted as apressure. As in [14,15], for simplicity we assume the homogeneous Dirichlet boundary condition for the velocity u . We consider the case of quasi-Newtonian flow and assume that the viscosity of the fluid obeys the Carreau law

where qo is the zero-shear rate viscosity, qw is the infinite-shear rate viscosity, 70 > qw > 0.The parameter X is a positive constant. Moreover, we assume r E ( 1 , 2 ] . The value r = 2 leads to the Newtonian flow, and detailed discussion of its finite element approximation can be found in [63]. For r E ( 1 , 2 ) , the problem models a pseudo-plastic fluid. Other possible forms of q can be found in [22]. To present variational formulations of the boundary value problem (5.54)(5.57), we need to introduce some function spaces. Let

v = W,l(Wd, U = { v E V : div v = 0 a.e, in R),

We observe that from the Korn's inequality (1.42), the formula

defines a norm on V . Assume the applied body force f E ( L ~ ( R )Then ) ~ .a variational formulation for (5.54)-(5.57) is to find u E V and p E M such that

B ( u ; u , v )- (p,divv) = ( f , v ) 'v'v E V, 'v'qM ~ . (div u , q ) = 0

(5.59) (5.60)

Here, (., .) is the L2( R )or (L2( R ) ) dinner product, and for w , u , v E V , 0

B ( w ;u , v ) = 2 JR

r l ( l ~ ( w ) 1~2() u. e)( v )dz.

The problem (5.59)-(5.60) is called a mixed formulation, because both the primal variable u and the dual variable p are present. Alternatively, we may eliminate the variable p to get a formulation involving the variable u only:

u E U,

B ( u ; u , v )= ( f , v ) 'v'v E U.

(5.62)

222


This problem is called the direct formulation in [15], and is equivalent to a constrained minimization problem

with

We notice that

( A 1 ( u v) ,) = B ( u ;u , v ) 'v'u,v E V. The mixed formulation (5.59)-(5.60) is the basis for developing mixed finite element methods. Error analysis for mixed finite element methods is done in [14] for the case r E ( 1 , 2 ] ,and in [15] for any r > 1,where improvements are made on some error estimates proved in [14]. , resulting fiBecause of the nonlinearity introduced by the function q ( - ) the nite element system is nonlinear and is difficult to solve. One possible approach is to apply the KaEanov method to reduce the nonlinear system to a sequence of linearized problems. We may analyze the KaEanov method on both the continuous level when it is applied to the original problem and the discrete level when it is applied to solving finite element systems. Here we describe the method on the discrete level. Let Vh C V and Mh c ~ i ( f lbe) finite dimensional spaces. Define Uh = { V h E Vh : (divvh,qh) = 0 'v'qh E Mh). We now consider the approximation of the problem (5.62):

The KaEanov method for solving the problem (5.65) is the following. Choose uo E Uh as an initial guess. Then for L = 1 , 2 , . . . , we compute uk E Uh from

Let us apply Theorem 5.5 to conclude the convergence of the KaEanov method (5.66) when it is applied to solve the problem (5.65). It is easy to show that B , as defined in (5.61), satisfies (5.41) and (5.42), where we may let Sl = 70and So = qm. From Lemma 3.1 of [15], the functional A defined in (5.64) satisfies (5.43), where we may take po = vm. In order to show the convergence of {uk) defined in (5.66), it suffices to verify the key inequality (5.44).


For q ( z ) defined in (5.58), we have

Hence, for any v , w E V ,

and the key inequality (5.36) is satisfied. Thus, the method (5.66) converges following Theorem 5.5.

A posteriori error estimate. Define the operator

where

Q

{ q = (qij) : qij = qji E L ~ ( R )1, 5 i, j We identify Q* with Q. Then we define the functional

< d)

with F

Obviously, E ( v ) = J ( v , A v ) . We now compute the conjugate functional J* (A*q*,-q* ) :

where t = t(lq*1) is determined by the algebraic equation

224


Since the function of t defined by the left-hand side of (5.67) is continuous, strictly increasing and taking on any non-negative value, for and q* E sd,the equation (5.67)has a unique solution t = t(lq*1 ) . Write ko = ko( jq* I ) = t(lq*I)/lq*1. Then ko is the solution of the equation

and

2 Xr

+ X k i iq*2)r/2

- - (70 - qm) [(1

-

1 ] }dx.

(5.69)

Define the constraint set

Then from (5.69),we get

14*12[(1- 1 1 ~ko) 2

J* (A*q*,-q*) =

+o

+m

-0

+ (217-- 1) r1&1 ifq* E

Qf,

(5.70)

otherwise.

Consider the quantity D ( u h ,u k )given by (2.21). From the definition of G ( q ) ,we have

and

Since r

Thus.

> 1 and ( p - q ) 2 5 lp121q12,

we have


Applying Theorem 2.40, we then get

Comparing the relations (5.66) and (5.70), we see that a convenient choice is q* = -2 q ( l ~ ( ~ k - l ) I 2 ) & ( u k )With . such a choice we get from (5.71) that an upper bound for both q,lluk - uh1I2and D ( u h ,u k )is

where ko is uniquely determined from (5.68) with

Notice that the expression (5.72) is guaranteed to be an upper bound for the left hand side of (5.71). It is computable once two consecutive iterates u k - ~and u k are found. Notice that usually numerical integration is needed to compute the expression (5.72).

E X A M P L E5.11 Let d = 2, R = [O, 112,f (x)= ( x 2 ,1 - x ~ )X ~= 2, , and r = 1.2. We use the Q1-Po element (see [55] for details). The domain R is divided into 9 equal squares. For the finite element spaces, Mh consists of piecewise constant functions and Vh consists of continuous vector-valued functions whose components are piecewise bilinear. A Gauss-Legendre quadrature of high degree is used to produce accurate values of Eh,k in (5.72). We use the effectivity index

to measure the efficiency of the error estimate. Some numerical results are given in Tables 5.11 and 5.12. We observe that the method converges very fast and the effectivity index Ok is close to 1, indicating that our a posteriori error estimate is effective. I


Table 5.11. Example 5.11, KaEanov iteration for a quasi-Newtonian flow problem, = 0.0001

770

= 1,

7700

Table 5.12. Example 5.11, Kacanov iteration for a quasi-Newtonian flow problem, 770 = 100, = 0.01

77-

5.5.

APPLICATION IN SOLVING AN ELASTOPLASTICITY PROBLEM

In this section, we apply the KaEanov method and the regularization method to solve a nonlinear variational inequality of the second kind arising in elastoplasticity, and derive a posteriori error estimates. The problem is discussed in detail in [130, 771. Here, we first briefly review one mathematical formulation of the problem. Consider the quasistatic behavior of an elastoplastic body which occupies a bounded domain R c IRd (d 5 3 in practice) with Lipschitz boundary. The plastic behavior of the material is described in terms of a dissipation function, and we assume the material undergoes nonlinear kinematic hardening, the nonlinear part of which takes the form of an exponential decay. The boundary value problem we are going to present arises in a typical time-step in approximating the rate of change of the plastic strain by a backward Euler difference (cf. [130]). We assume the material is subject to the action of a body force with density b. For simplicity in presentation, we assume the boundary of the body is fixed. The unknown variables of the problem are the displacement u and the plastic strain tensor p. We seek the displacement in the space

and the plastic strain in the space

Qo = {q E Q : trq =0},


where t r p = pii is the trace of the tensor p , and

We require the restriction t r p = 0 on the plastic strain p by the conventional assumption of no volume change accompanying the plastic deformation. Both V and Q are Hilbert spaces with inner products

and the corresponding norms 11 v 11 = ( v ,v )' I 2 , 11 q 11 Q = ( q ,q ) 1 / 2 . Moreover, Qo is a closed subspace of Q. = V x Q To formulate the problem, we need to use the product space which is a Hilbert space with the inner product

v

and norm (/Ellt; = (8,8 ) g 2for , 8 = (u,p),B = ( v ,q ) E V . We also define V o = V x Qo, a closed subspace of Define an operator A : V + by

-

v*

v.

where

and

h ( a ) = ho

+ hie-pa.

We assume the fourth-order elasticity tensor (C to be symmetric:

bounded:

CijklE L m ( R ) , 1 5 i , j , k , l 5 d , and pointwise stable: for some constant co > 0,

We also assume that the coefficient p the conditions

> 0 and the functions ho and hl

satisfy

228


The smallness assumption on hl (relative to ho) is needed in proving the unique solvability of the elastoplasticity problem (5.74) below (cf. [130]). Then, associated with the density of body force b E V* we define the linear functional

e : V -+ R, e ( q = and for some material parameter g

S, .

b v dx

> 0 we define the functional

The functional j ( . ) is known as the dissipation function in plasticity. The functionals e ( . ) and j ( . ) are easily shown to be bounded, j ( . ) is a convex, positively homogeneous (i.e., j(XZI) = X j ( G ) for any X > O), non-negative Lipschitz continuous functional. Note that, however, j is not differentiable. The variational form of the elastoplasticity problem is: Find 2i = ( u ,p ) E Vo such that

By applying Theorem 1.25, one can show that the problem has a unique solution (see also [130]).

The Karanov method and its convergence. The KaEanov method for solving the problem (5.74) is: Choose an initial guess Go E T o ; for k = 1 , 2 , . . . , find u k E VOsuch that

where

To apply Theorem 5.5 for a convergence analysis of the KaEanov method, we notice that the problem (5.74) is equivalent to the minimization problem of finding U = ( u ,p ) E Vo such that

where

E ( v )= E l @ )

+ j ( F ) - l(5)


229

and

A crucial step in applying Theorem 5.5 for the convergence of the method (5.75) is to prove the key inequality (5.36), which in the present case, is equivalent to (5.78) in the next lemma.

LEMMA5.12 The following inequality holds:

Proof. For a fixed s

> 0, we define a function o f t by

Then from the definitions of H (.) and h ( . ) ,

Since f ' ( t )= p2t ( e - p t - e - p S ) ,

>

>

we have f ' ( t ) 0 for t E [0, s] and f 1 ( t )5 0 for t s. We conclude that the function f ( t ) ,t 2 0, attains its maximum value at t = s. Thus,

f ( t )5 f(s) = o

'dt > 0.

Hence, the inequality (5.78) is proved. The other conditions of Theorem 5.5 are easily verified. Therefore, the KaEanov method (5.75) converges.

A posteriori error estimation for the KaEanov method. Now that the convergence of the KaEanov method (5.75) is shown for the problem (5.74), we turn to derive some a posteriori error estimate for the KaEanov iterates. To derive an a posteriori error estimate for the approximation error (uk- u), we apply Theorem 2.40. Since it is customary to use p for the plastic strain, we will use s for a generic dual variable, and S for the space of the dual variables. Define the operator A : V o-+ S, Av = ~ ( v ) ,

230


We identify S* with S . Now define the functional

Then, the problem (5.76) can be rewritten as

J ( E , A E ) = inf { J ( B , A B ):;is E T o ) .

E E V o:

(5.79)

Note that the functional J(W, s ) is not of separated form. Define a constraint set

S; = { s * E S *

[E(v)-s*+b.v]dx=O V v EV

We can find the conjugate functional

J*(A*s*,-s*) =

{

[ic-ls* OC,

S*

I

+ K ( S * I ) dx

ifs* t

s.,

otherwise.

Here, ~ s* - ( l l d )tr ( s * ) I ,I being with the deviatoric strain tensor defined as s * = the second-order identity tensor,

If I S * ~ 5 ~ g, then t ( l ~ * = ~ I0.) If solution of the equation

> g, then t ( l ~ * ~> I0)is the unique

The unique solvability is guaranteed by the assumption ho > 0. We will say an s* E S* is admissible if s* E S,*. Notice that from (5.80), the value of J* (A*s*,- s * ) is infinite if s* is not admissible. Since the functional J ( B , s ) is not Giteaux differentiable, we consider the energy difference

for some constant E

> 0 (for an expression to calculate E > 0, see [77]).


23 1

To have an upper bound for the difference, we need to choose a suitable admissible variable s*. We notice that the problem (5.75) is equivalent to

and

From (5.83), it follows that

s* = -C ( E ( u k )- p k ) is admissible. With this choice of s * , applying Theorem 2.40, we obtain the following upper bound on E D @ , T i k ):

Applying (5.83) with v = u k , we obtain

+ H ( l ~ k l )+ g

pk

+ T ( ~ ( ~ ~ * ~ I ) ) ] dx.

Combined with (5.82), this implies the following a posteriori error estimate ( I u k - ~112, + lpk -PI/;)

where s * is defined in (5.85). For the purpose of comparing with the a posteriori error estimate to be derived next for the combined effect of a regularization procedure and the KaEanov iteration, we derive another a posteriori error estimate for the KaEanov method (5.75) as a consequence of the estimate (5.86). We take q = 0 in (5.84) to obtain

232


Applying this inequality to the right-hand side of (5.86), we get the estimate

Regularization, a posteriori error estimate. We observe that in each step of the KaEanov iteration, one has a (linear) variational inequality of the second kind, (5.75). The difficulty in solving (5.75) directly lies in the fact that term j is non-differentiable. One approach used in practice for overcoming this difficulty is to use a regularization technique. In a regularization technique, j is replaced by a differentiable function j,, such that j, + j as E + 0. As is discussed in Section 5.1, there are many choices for the regularization function j,. Here, for definiteness, we take

Once the regularization function j, is chosen, the variational inequality (5.75) is approximated by the following

Owing to the fact that j, is differentiable, the regularized problem can be rewritten as B ( T i k - l ; T i E , k , C ) + ( j ~ ( ~ , , k ) , ~ 'dvEV0, )=e(v) or, in detail,

The convergence of the regularization method can be established by a standard procedure. From (5.16), we have some constant c such that

+ l l p k - p E , k I ; 5 C&.

l l ~ k- ~ & , k l l $

We now derive a posteriori error estimates for the approximation error Ti). Again we consider the energy difference

-

233


It can be proved that

for the same constant Zi > 0 as in (5.82). By (5.881, S*

= - c ( & ( ~ & , k -) P E , ~ )

is admissible. With this choice of s * , we have

Combined with (5.89), this implies the following a posteriori error estimate

] dx.

(5.91)

To see the efficiency of the estimate, we have that in the linear case (i.e., hl = O), the estimate (5.91) implies

g l ~ e , 1k&2

'/ JW(JW+ dx.

IP&,~I)

It is easy to see that, at least formally, the error estimate (5.91) reduces to (5.87) when E + 0.

Chapter 6

ERROR ANALYSIS FOR VARIATIONAL INEQUALITIES OF THE SECOND KIND

The finite element method today is the dominant numerical method for solving most problems in structural and fluid mechanics. It is widely applied to both linear and nonlinear problems. For practical use of the method, one of the most important problems is the assessment of the reliability of a finite element solution. The reliability of the numerical solution hinges on our ability to estimate errors after the solution is computed; such an error analysis is called a posteriori error analysis. A posteriori error estimates provide quantitative information on the accuracy of the solution and are the basis for the development of automatic, adaptive procedures for engineering applications of the finite element method. The research on a posteriori error estimation and adaptive mesh refinement for the finite element method began in the late 1970's. The pioneering work on the topic was done in [l 1, 121. Since then, a posteriori error analysis and adaptive computation in the finite element method have attracted many researchers, and a variety of different a posteriori error estimates have been proposed and analyzed. In a typical a posteriori error analysis, after a finite element solution is computed, the solution is used to compute element error indicators and an error estimator. The element error indicator represents the contribution of the element to the error in the computation of some quantity by the finite element solution, and is used to indicate if the element needs to be refined in the next adaptive step. The error estimator provides an estimate of the error in the computation of the quantity of the finite element solution, and thus can be used as a stopping criterion for the adaptive procedure. Often, the error estimator is computed as an aggregation of the element error indicators, and one usually only speaks of error estimators. Most error estimators can be classified into residual type, where various residual quantities (residual of the equation, residual from derivative discontinuity, residual of material constitutive laws, etc.) are used; and recovery type, where a recovery operator is applied to the (discontinuous)

236


gradient of the finite element solution and the difference of the two is used to assess the error. Error estimators have also been derived based on the use of hierarchic bases or equilibrated residual. Two desirable properties of an a posteriori error estimator are the reliability and efficiency. The reliability requires the actual error to be bounded by a constant multiple of the error estimator, up to perhaps a higher order term, so that the error estimator provides a reliable error bound. The efficiency requires the error estimator to be bounded by a constant multiple of the actual error, again perhaps up to a higher order term, so that the actual error is not over-estimated by the error estimator. The study and applications of a posteriori error analysis is a current active research area, and the related publications grow fast. Some comprehensive summary accounts can be found, in chronicle order, in [153], [2], and [13]. Initially, a posteriori error estimates were mainly developed for estimating the finite element error in the energy norm. In the recent years, error estimators have also been developed for goal-oriented adaptivity. The goal-oriented error estimators are derived to specifically estimate errors in quantities of interest, other than the energy norm errors. Chapter 8 of [2] is devoted to such error estimators. The latest development in this direction is depicted in [19, 621. Most of the work so far on a posteriori error analysis has been devoted to ordinary boundary value problems of partial differential equations. In applications, an important family of nonlinear boundary value and initial-boundary value problems is that associated with variational inequalities. Although several standard techniques have been developed to derive and analyze a posteriori error estimates for finite element solutions to problems in the form of variational equations, they do not work directly for a posteriori error analysis of numerical solutions to variational inequalities. Nevertheless, numerous papers can be found on a posteriori error estimation of finite element solutions of obstacle problems, e.g., [3], [40], [84], [98], [I 141, [I501 (these papers consider numerical solutions on convex subsets of finite element spaces), as well as [56], [90] (these papers use a penalty approach for discrete solutions). Obstacle problems are so-called variational inequalities of the first kind, that is, they are inequalities involving smooth functionals and are posed over convex subsets. We also note that a posteriori error estimation is discussed in [24, 25, 1451, though the arguments in these papers are arguable. In the context of elastoplasticity with hardening, computable a posteriori error estimates are derived in [4, 34, 361 for the primal problem, which is a variational inequality of the second kind; that is, the inequality arises as a result of the presence of a non-differentiable functional. These works deal extensively also with a priori estimates, and in the latter work a number of numerical examples are presented. Residual type error estimators were studied for an elliptic variational inequality of the second kind in 128, 291.

Error Analysis for Variational Inequalities of the Second Kind

237

In this chapter, we derive and study some a posteriori error estimates for finite element solutions of elliptic variational inequalities of the second kind. The basic mathematical tool we will use is the duality theory in convex analysis, although some of the a posteriori error estimates can be derived by more direct calculations. In [134,133], the technique of the duality theory was used to derive a posteriori error estimates of the finite element method in solving boundary value problems of some nonlinear equations. In these papers, the error bounds are shown to converge to zero in the limit; however, no efficiency analysis of the estimates is given. In Section 6.1 we introduce a model elliptic variational inequality of the second kind and its finite element approximation. We will provide detailed derivation and analysis of a posteriori error estimates of the finite element solutions for the model problem. In Section 6.2 we formulate a dual problem for the model, and use the dual problem to establish a general a posteriori error estimate for any approximation of the solution of the model problem. The general a posteriori error estimate is featured by the presence of a dual variable. Different a posteriori error estimates can be obtained with different choices of the dual variable. In Section 6.3, we make a particular choice of the dual variable that leads to a residual-based error estimate of the finite element solution of the model problem, and explore the efficiency of the error estimate. In Section 6.4, we make another choice of the dual variable and obtain a gradient recoverybased error estimate of the finite element solution of the model problem. We also study the efficiency of the error estimator. In Section 6.5 we present some numerical results to illustrate the effectivity of the estimates in adaptive solution of the model variational inequality. In the last section of the chapter, we extend the discussion to solving a frictional contact problem. We comment that the discussions made here can be extended in solving time dependent variational inequalities, cf. e.g. [27].

6.1.

MODEL PROBLEM AND ITS FINITE ELEMENT APPROXIMATION

We introduce a model elliptic variational inequality of the second kind in this section. We comment that the ideas and techniques presented for a posteriori error analysis in solving the model problem can be extended to other elliptic variational inequalities of the second kind; in particular, in Section 6.6 we provide a posteriori error analysis for the finite element solution of a frictional contact problem. Let R be a domain in E t d , d 2 1, with a Lipschitz boundary I?. Let Fl c r be a relatively closed subset of r , and denote F2 = r\rl for the remaining part of the boundary. We allow the extreme situations where rl = 0 (i.e, r2= r ) and rl = r (i.e. r2= 0). Since the boundary is Lipschitz continuous, the unit outward normal vector Y exists a.e. on r . We will use d l d u to denote

238


the outward normal differentiation operator, that exists a.e. on I?. f E L~(R) and g > 0 are given. Over the space

Assume

we define a bilinear form and two functionals: a ( u , v) =

( V u - Vv

S,

+ u v) dx,

In the space V, we use the H1(R)-norm. The model problem is the following elliptic variational inequality of the second kind: Find u E V such that

This model is a so-called simplified friction problem following [65] as it can be viewed as a simplified version of a frictional contact problem in linearized elasticity (cf. Section 6.6). Since the bilinear form a ( . , is continuous and V-elliptic, the linear functional e(.) is continuous, and the functional j ( - ) is proper, convex and continuous, by Theorem 1.25, the variational inequality (6.1) has a unique solution. Moreover, due to the symmetry of the bilinear form a ( . , .), the variational inequality (6.1) is equivalent to the minimization problem: Find u E V such that E ( u ) = inf E ( v ) , (6.2) a )

vEV

where E is the energy functional:

The minimization problem (6.2) also has a unique solution. In the analysis of a posteriori error estimators later, we will need the following characterization of the solution u of (6.1): There exists a unique X E LCO(rz)such that

The function X can be viewed as a Lagrange multiplier.


239

A proof of this characterization in the case r2= I'can be found in [65]. The argument there can be extended straightforward to the more general situation considered here; see also the proof of Theorem 6.2. It follows from (6.4) that the solution u of (6.1) is the weak solution of the boundary value problem

We now turn to finite element approximations of the model problem. For simplicity, we suppose that R has a polyhedral boundary I'.In order to define the finite element method for (6.1) we introduce a family of finite element spaces Vh c V, which consist of continuous piecewise polynomials of degree larger than or equal to 1, corresponding to partitions Phof into triangular or tetrahedral elements (other kinds of elements, such as quadrilateral elements, or hexahedral or pentahedral elements, can be considered as well). The partitions Phare compatible with the decomposition of r into rl and r2.In other words, if an element side lies on the boundary, then it belongs to one of the sets I'l or r2.For every element K E P h , let hK be the diameter of K and p~ be the diameter of the largest ball inscribed in K . For a side y of the element K , we denote by h, the diameter of y. We shall assume that the family of partitions p h , h > 0, satisfies the shape regularity assumption, i.e. the ratio h K / p K is uniformly bounded over the whole family by a constant C. Note that the shape regularity assumption does not require that the elements be of comparable size and thus locally refined meshes are allowed. We will use Eh for the set of the element sides, Eh,r, Eh,rl, and Eh,rz for the subsets of the element sides lying ~ the subset of the element on I?, r l , and E,respectively, and Eh,0 = E h \ E h ,for sides that do not lie on r. Let Nh be the set of all nodes in Ph and Nh,0c Nh the set of free nodes, i.e. those nodes that do not lie on rl. For a given element K E P h , N(K) and & ( K ) denote the sets of the nodes of K and sides of K , respectively. The patch g associated with any element K from a partition Ph consists of all elements sharing at least one vertex with K , i.e, g = U{Kt E Ph : Kt n K # 8). Similarly, for any side y E Eh, the patch 7 consists of the elements sharing y as a common side. Note that in the case where the side y lies on the boundary r , the patch 7 consists of only one element. For a given element K E P h , U K denotes the unit outward normal vector to the sides of K. When a side y lies on the boundary I?, v , denotes the unit outward normal vector to I?. For a side y in the interior, v , is taken to be one of the two unit normal vectors. In what follows, for any piecewise continuous function cp and any interior side y E Eh,O,[cp], denotes the jump of cp across y in the direction


v,, i.e.

In the derivation of a posteriori error estimates we will use the so-called weighted ClCment-type interpolation operator. There are several variants of such operators (see e.g. [16], [35], [37], [38], [158]), which are all versions of the interpolation operator introduced by Clkment [44]. The main difference among these interpolants lies in the way the interpolation is performed near the boundary. In this paper we will follow the approach used in [37]. Corresponding to the partition P h , we denote N j C Nh to be the set of the element vertices, Nu,rlc N;, the subset of the element vertices lying on r l , = N;, n Nh,0the subset of the interior vertices. Given a E N u , let and 4, be the linear element nodal basis function associated with a. For each fixed vertex a E Nu,rl,choose ((a) E Nuloto be an interior vertex of an element containing a . Let ( ( a ) = a if a E NuloFor each node a E Nu,odefine the class I(a) = {a E Nu : ((a) = a). In this way, the set of all the vertices N,, is partitioned into card(Nulo)classes of equivalence. For each a E Nuloset

Notice that {ga : a E N u , o v sa partition of unity. Let Fa = supp($,) and ha = diam(&). The set Ka is connected and that $, # Pa implies that I'l n & has a positive surface measure. , For a given v E L ~ ( R )let

Then define the interpolation operator Uh : V

--+ Vh as follows:

The next result summarizes some basic estimates for Uh. Its proof can be found in [35].


THEOREM 6.1 There exists an h-independent constant C all v E V and f E L2(0),

24 1

> 0 such that for

The finite element method for the variational inequality (6.1) reads: Find

uh E Vhsuch that

The discrete problem has a unique solution uh E Vh. We need the following characterization of the finite element solution, similar to that of the solution of the continuous problem.

THEOREM 6.2 The unique solution uh E Vhof the discrete problem (6.11) is characterized by the existence of Ah E Lw (r2)such that

Proof. Assuming (6.1I), let us prove (6.12) and (6.13). Taking first vh = 0 and then vh = 2uhin (6.1 I), we obtain

Together with (6.14) the relation (6.11) leads to

v:,

HA

where V; = Vhn (0) and : V is the orthogonal Write Vh = V; @ complement of Vhin H : ~(0). It follows from (6.15) that !(vh) - a(uh, vh) = 0 V vh E Notice that the trace operator from V; onto : V lrz c L1(rz)is

v;.

242


an isomorphism. Therefore, the mapping L ( v h ) = t ( G h ) - a(uh,Gh) can be viewed as a linear functional on v:/~, , where Zh is any element from the space Vh whose trace on F 2 is vh. It follows from (6.15) that

Thus, by the Hahn-Banach Theorem the functional L(vh) can be extended to L ( v ) on L 1 ( r 2 )and so there exists Ah E Lm(F2)such that

and lXhl 5 1 a.e. on F2, from which (6.12) follows. Taking now vh = uh in relation (6.12), we have

and using (6.14) we get

Since / A h I 5 1 a.e. on r 2 ,we must have luh I = Xhuh a.e. on r2.This completes the proof of (6.12) and (6.13). Conversely, assume (6.12) and (6.13) hold. It follows from relation (6.12) that

which can be rewritten as

Then, relation (6.13) implies that

Since Xhvh 5 )vh)a.e. on r 2 ,it follows immediately that uh is the solution of the discrete problem (6.11). A priori error estimates for the finite element method (6.11) can be found in the literature, e.g. [65, 661. Here, we focus on the derivation and analysis of a


243

posteriori error estimators that can be used in adaptive finite element solution of variational inequalities. Our argument of the efficiency of the a posteriori error estimators is based on that due to Verfiirth [153], with special attention paid to the inequality feature of the problem. The argument makes use of the canonical bubble functions constructed for each element K E Phand each side y E & ( K ) . Denote by PK a polynomial space associated with the element K . The following two theorems provide some basic properties of the bubble functions used to derive lower bounds. For more details on bubble functions and proofs see [2].

THEOREM 6.3 Let K E Phand gK be its corresponding bubble function. Then there exists a constant C, independent of hK, such that for any v E PK, the following inequalities hold:

THEOREM 6.4 Let K E Phand y E & ( K )be one of its sides. Let gY be the side bubble function corresponding to y. Then there exists a constant C, independent of h K , such that for any v E PK,

6.2.

DUAL FORMULATION AND A POSTERIORI ERROR ESTIMATION

We now present a dual formulation for the model problem. The dual formulation will be used in the derivation of a posteriori error estimators for approximate solutions. Let Q = ( L ~ x L~(a)x L2(r2). Define a function J : V x Q + by the formula

where q = (ql, q 2 , q3) E Q. Introduce a linear bounded operator A : V by the relation b'~ E V. AV = ( V V v,vlr2) , Then obviously, E ( v ) = J ( v , A v ) b'v E V.

+Q


244

Therefore, the minimization problem (6.2) can be rewritten as: Find u E V such that (6.17) J(u,Au) = inf J(v,Av). vEV

Let V * and Q* = ( L(~R ) )x~L~( 0 )x L~(r2)be the duals of V and Q placed in duality by the pairings (-, .) and (., a ) Q ,respectively. Let us compute the conjugate function

where A* : Q* + V * is the adjoint of A. We have

j*(A*q*, -q*) = sup

v v + (q;+ f) v ] dx +

[ q .~

vEV q€Q

It can easily be verified that

0, if lq;/ 5 g a.e. on r2, oo,

otherwise.

Also, note that the term sup UEV

equals

CG, unless

{S,

[q;. Vv

q* E

+ (qi+ f ) vl dx +

S,,

q:v ds}

Q*satisfies

and under this assumption, the above term equals 0. Define the set of admissible dual functions


The classical relations for the constraint q* = (q?,q;, qg) E Qj,g are

Then the conjugate function (6.18) is

J * ( A * ~ *-q*) , =

Jn

Qi,gj

(q;12+ 1$12)dx, i f q * E otherwise.

The dual problem of (6.17) can now be stated: Find p* E Q j,gsuch that - J*(A*p*,-p*) =

{ - J*(h*q*,- q * ) ) .

sup q*

(6.20)

EQj.,g

Note that the mapping q* t+ J*(A*q*,-q*) is strictly convex over Qj,g. By Theorem 2.39, the dual problem (6.20) has a unique solution p* E Q jig, and

where u E V is the unique solution of the model problem. Now let w E V be an (arbitrary) approximation of u E V, the unique solution of (6.1). In the rest of the section, we present a general framework for a posteriori estimates of the error ( u - w ) . The error bounds are computable from the (known) approximant w. Later on, w will be taken as the finite element solution of the variational inequality. By using (6.1) and (6.3) we obtain

On the other hand, let p* be the solution of the dual problem (6.20). Relation (6.21) implies

E ( u ) = J ( u , Au) = - J* (A*p*,-p*) Therefore,

> -J*(A*q*,-q*)

V q* E Qi,g.

246


Introduce the space

Q; = ( L ~ ( Qx) ~ ) ~~ ( 0 ) . Then for any r* = ( r ? r;) , E Q;, we write

Since q* = ( q ; ,qz, qi) E

Q>,g,

from (6.19),

Using this relation in (6.23) and recalling (6.22), we find that

Thus, we have established the following result.

247


T H E O R E6M . 5 Let u E V be the unique solution of ( 6 . 1 ) , and w E V an approximation of u. Then the following estimate holds for any r* E Q::

Let us now deal with the second term I1 on the right side of the estimate (6.24). First, from the definition (6.19) it follows that

+ lq; - r z 2 )dx +

= inf sup { L ( l q ;- r;I2 EQ* V E V 14; 159

q*

+

( g /wl q j w ) ds

Here and below, the condition ''/q;l 5 g" stands for ''/q;I 5 g a.e. on Substitute q ; - ry by qT, qz - rz by q; and regroup the terms to get

Define the residual

r2".


248 Then

+

= inf ( ~ ( 9 3r *, ) 2

Jr2

(9 I W

+ d w )d

~ )

14;/5g

This last estimate is combined with Theorem 6.5, leading to the next result.

T H E O R E6.6 M Let u E V be the unique solution of (6.1), and w E V an approximation of u. Then the following error bound

+

Iq;15g inf

+

{ ~ ( q ; , r * ) 1~2 ( g l w l+ q j w ) d s } (6.26)

'v'r*= ( r T , r $ )E Q: is valid, where the residual R ( q $ ,r * ) is defined by (6.25).

6.3.

RESIDUAL-BASED ERROR ESTIMATES FOR THE MODEL PROBLEM

With the special selection rT = - V w , rz = -w, (6.26) leads to the following error estimate: I,

\/;a(u--W,U-w)

< R,

(6.27)

where

Although it is possible to derive (6.27)-(6.28) through other approaches, we comment that Theorem 6.6 provides a general framework for various a posteriori error estimates with different choices of the auxiliary variable r*. In the limiting case g = 0, the problem (6.1) reduces to the variational equation U E V , a ( u , v ) = l ( v ) 'V'vEV. We observe that correspondingly, the estimate (6.27)-(6.28) reduces to the familiar form

/:

-a(u

-w,u -w)

J

5 sup v€V

I/vI/~

R

[(f

- W)V

- vw

. VVI dx,


249

which is a starting point in deriving some a posteriori error estimators for Galerkin approximations of linear elliptic partial differential equations (cf. [2]). Now we focus on a posteriori analysis for the finite element solution error; that is, for the situation where the approximant w = uh is the finite element solution. By taking qi = -g Ah, and substituting v by vh - v in (6.28) we obtain

{L

1 R 5 sup V E VIIvllv

.

[ ~ u hV ( V

- vh) + (uh - f) ( v - vh)I dx

for any vh E Vh.Here we have used (6.12). For a given v E V, take vh = Dhv in (6.29), where Dhv is defined in (6.6). Decompose the integrals into local contributions from each element K E Ph and integrate by parts over K to obtain

Define interior residuals for each element K E Ph by

and side residuals for each side y E &h,0,r2

-- &h,O U Eh,J2 by

where the quantity

represents the jump discontinuity in the approximation to the normal derivative on the side y which separates the neighboring elements K and K t . By using


250

definitions (6.31) and (6.32), relation (6.30) reduces to

Using the estimates (6.9) and (6.10) in (6.33), and applying the CauchySchwarz inequality, we have

R 5 sup uEV llvllv

c lv11,n(

+

~ $ I I ~ K I I ~ , K hT1lR&Y) K E P ~ ?'EEh,o,rz

'

We summarize the above results in the form of a theorem.

THEOREM 6.7 Let u and respectively. Then the error

uh

be the unique solutions of (6.1) and (6.11), - uh satisjes the a posteriori estimate

eh = u

where r~ and R, are interior and side residuals, defined by (6.31) and (6.32), respectively. In practical computations, the terms on the right side of (6.35) are regrouped by writing

lehI:.

5 CV;,

~a

=

c +iK1

where the local error indicator V R , K on each element K, defined by

identifies contributions from each of the elements to the global error.

25 1


In the last part of the section, we explore the efficiency of the error bound from (6.35) or (6.36). We derive an upper bound for the error estimator. Integrating by parts over each element and using (6.4) and Theorem 6.13 we have, for any v E V,

( A- uh + j)v dx + KEP,

YE&~,O

-

[g] Y

v ds

Thus,

where 7-K and R, are interior and side residuals defined for each element K E side y E &h,O,rzby (6.31) and (6.32), respectively. In order to simplify notation, we will omit the subscripts K and y. In the following, we will apply Theorems 6.3 and 6.4 choosing PK as the space of polynomials of degree less than or equal to 1, and 1 is any integer larger than or equal to the local polynomial degree of the finite element functions. Let T be a discontinuous piecewise polynomial approximation to the residual r , that is, TI E PK. Applying Theorem 6.3, we get

Phand each

Since the function v = $KT vanishes on the boundary aK,it can be extended to a function in V by 0 to the rest of the domain R. Inserting this extended function v in the residual equation (6.38), one obtains

Using this relation, we obtain


The terms on the right side of (6.40) are bounded by making use of the CauchySchwarz inequality and the second part of Theorem 6.3,

Combined with (6.39) we have

With the aid of the triangle inequality, finally we get

Consider now an interior side y E it follows that

IlRlI;,,

&h,O. From

IC

I

the first part of Theorem 6.4

$,R2ds.

(6.42)

Let 7denote the subdomain of fl consisting of the side y and the two neighbouring elements. The function v = $,R vanishes on 87 and as before it can be extended continuously to the whole domain R by 0 outside 7.With this choice of v the residual equation (6.38) reduces to

Therefore,

0

Applying the Cauchy-Schwarz inequality and the second part of Theorem 6.4 to the terms on the right side of (6.43), we obtain

which, combined with (6.42) and (6.41), implies that for every interior side

E &h,03 1lRlIo.i

1/2 -

( h ~ 1 i 2 ~ ~ e+hh7 ~ ~ \Iri , i-. r ~ ~ o ,. ? )

(6.44)

Finally, consider those sides y lying on E. Denote R E PK an approximagXh on y, y E E (K). The first part of Theorem tion to the residual R = 6.4 implies

%+


253

and let y be the element whose boundary contains Define the function v = the side y.Then = 0. Extend this function to the whole domain by zero value outside 7. The residual equation (6.38), with this choice of v , becomes

~

l

~

~

\

~

which leads to

As before, the first three terms on the right side of (6.46) can be bounded by applying Theorem 6.4 and the Cauchy-Schwarz inequality. Using (6.45), we then obtain for each side y E & h , r z ,

Multiplying this inequality by h7 and summing over all sides y E Ehlr2, we get

where

We can bound Rh,r2as follows,

254


Use this bound in (6.47) and apply the Cauchy-Schwarz inequality to get

Combining (6.41) and (6.48), we finally conclude that

Summarizing (6.41), (6.44) and (6.49), we have

For our model problem, the element residual r K of (6.31) and the side residual Ry of (6.32), -nuh uh in K and d u h / d v on y are polynomials. Therefore, the terms llrK - - I O , K and 11 R, - %110, , in the right hand side of (6.50) can be replaced by 11 f - f K 1 1 0 , ~ and 11 Ah - Ah,? 1 0 , , with discontinuous piecewise polynomial approximations f K and Ah,y.

+

T H E O R E6.8 M Let 7~ be dejined as in (6.36). Then

with discontinuous piecewise polynomial approximations f K , Ah,y o f f , Ah.

Error Analysis for Variational inequalities of the Second Kind

255

Let us comment on the three summation terms in (6.51). As long as f has suitable degree of smoothness, the approximation error CKtPh hgll f f K / I f , , will be of higher order than 1 eh I & . Due to the inequality nature of the variational problem, in the efficiency bound (6.5 1) of the error estimator, there are extra terms involving X and Ah. A sharp bound of the term '&,h,r2 h, / X -

X h / & is currently an open problem. Nevertheless, in Section 6.5, we will present numerical results showing that the presence of this term in (6.51) does not have an effect on the efficiency of the error estimator. Similar numerical results can also be used as an evidence that the term CYeEh,r2 h-, l X h - X h j 7 1 f , , does not have an effect on the efficiency of the error estimator.

6.4.

GRADIENT RECOVERY-BASED ERROR ESTIMATES FOR THE MODEL PROBLEM

An important class of a posteriori error estimates is based on local or global averaging of the gradient, e.g, in the form of Zienkiewicz-Zhu gradient recovery technique, [165], [166], [167]. In [16], [37], Carstensen and Bartels proved that all averaging techniques provide us with a reliable a posteriori error control for the Laplace equation with mixed boundary conditions on unstructured grids as well. In the context of solving variational inequalities, gradient recovery type error estimates for elliptic obstacle problem have been derived recently in [17], [158]. In this section, we study a gradient recovery type error estimator for the finite element solution of the model problem, and we restrict our discussion to linear elements. In this case, the nodes are also the vertices. To formulate the error estimator we need a gradient recovery operator. There are many types of gradient recovery operators. In order to have a "good" approximation of the true gradient Vu, a set of conditions to be satisfied by the recovery operator were identified in [2]. These conditions lead to a more precise characterization of the form of the gradient recovery operator, summarized in Lemma 4.5 in [2]. In particular, the recovered gradient at a node a is a linear combination of the values of V u hin a patch surrounding a. We define the gradient recovery operator Gh: Vh + ( v ~ as follows: )~

Since linear elements are used,


256

where ( V v h )K i denotes the vector value of the gradient V v h On the element K;, Ka = K~,CX = ;I K ; I / K a , i = 1, . . . , Na. Recall from Section 6.3 that residual-type error estimates are derived by applying Theorem 6.6 with r* = - ( V u h , u h ) ,where uh is the finite element solution. In this section, we consider a different choice.

uF1

T H E O R E6.9 M Let u and uh be the unique solutions of (6.1) and (6.11), respectively. Then the error u - uh satisJies the a posteriori estimate

(6.54)

where

Proof. Let Ah E L m ( r 2 )be provided by Theorem 6.2. Apply Theorem 6.6 with w = uh and r* = -(Ghuh, u h )to obtain

where

1

{L

R = sup VEV

llullv

!Gnuh V r

+ (uh- f)u] dx +

/r2

ghhv ds}

.

(6.58)

Let Dh be the interpolation operator defined by (6.6). Use (6.12) with vh = nhv,

Therefore, we can rewrite (6.58) as

{ L [ G h ~.hV ( U nhv) + (uh - f )

1 R = sup V E V llullv

-

( U - nhu)l

dx


[n (Ghuh

1 sup VEV

IIvIIv

-

V u h ). V n h v dx 5 C l(Vuh- Ghuhlo,~.

where

Integrate by parts over each element K E Ph to get

R1 = sup uEV

/ / " / /KvE p h

+

{k(-diV(~hUh)

Uh -

f ) (V

- n h ~dx )

Since Ghuh is continuous, the integrals on the interior sides y E Eh,O cancel each other. Write

-div(Ghuh)

+Uh - f

+

= div(Vuh - Ghuh) (-Auh

+

Uh -

and rearrange the terms in (6.60) to obtain

Rl= SUP -

div(Vuh - Ghuh)(V

-

DhV) dx

f)

25 8


+

where r K = -Auh uh - f = uh - f denotes the interior residual on element K E Fh. We will use r to denote the piecewise interior residual, i.e. r 1 = r K for K E Ph. To estimate the first summand on the right side of (6.61), we use an elementwise inverse inequality of the form

Apply the Cauchy-Schwarz inequality, the inverse inequality (6.62) and the estimate (6.9) to get

5

c

(z

u

-

G h h K )l

2

KEP,

(

)

h i l ( v - nhv)lli,K

KEPh

For the second summand, we apply the estimate (6.8) to obtain

Now

C

h i min llr ra ER

~ E N ~ , o

2

- rallo,za

C

52

hilluh

-

~ I l i , ~

~ E N ~ , o

+2

x

h i min \ I f - fall2 -

faER ~ E N ~ , o

O,Ka t

where %denotes the integral mean of uh over Fa. Use the Poincare inequality and an inverse inequality of the form (6.62) to get


Therefore,

Finally, with the aid of the Cauchy-Schwarz inequality and the estimate (6.lo), the third summand on the right side of (6.61) can be bounded by

Inserting (6.63), (6.64) and (6.65) into (6.61), and using the Cauchy-Schwarz inequality and (6.59), we deduce that

Split the first term on the right side of estimate (6.57) into local contributions from each K E Ph and insert (6.66) to conclude the proof. We can write (6.54) as

where

We observe that usually the term Rh is of higher order compared to llu - uh 11 l,n which is of order O ( h ) in the nondegenerate situations. This observation is argued as follows: ~ i i s t it, is easy to show from the definition of the finite element solutions that there is a constant C such that IIuh 11 l,n C for any h. So the term

p q, where p is a prescribed threshold. In the example of this section, p = 0.5.

5 Perform refinement and obtain a new triangulation Ph. 6 Return to step 2. In the computation of the error indicator q ~ we make use of the multiplier Ah defined on r2c r. In what follows we describe how Ah can be (approximately) recovered from the solution uh using characterization (6.12). Denote by { a i ) z n the = l nodes of the partition Ph belonging to G, and let { 4 i ) z 1be the basis functions corresponding to the nodes { a i ) . We first determine a piecewise linear function

by requiring an analogue of (6.12):

264


Denote X h , ~= ( x ; , ~. ,. . , the interval [ - I , I ] to get Xk.2 =

X T ~ )We~ then . project the components of X h j l onto 1 = ( x ~ ., . ~ . , X, T ~ ) ~ :

max{min{~;,,, l ) ,-I),

i

= 1 , . . . , m.

The piecewise linear approximation of the multiplier Ah on computed as

r2 c r can be

m

We briefly comment on the method for finding Ah,,. Let n = dim Vh.Denote by K the standard ( n x n) stiffness matrix and by 1 E Rn the standard load vector. Let u E Rn be the nodal value vector of the finite element solution uh. Then the algebraic representation of (6.77) becomes

where vc denotes the subvector of v containing the nodal values of vh at the nodes { a i ) ~c, F 2 , and M is a tridiagonal ( m x m ) matrix. We can write v = ( v r ,v : ) ~E RnPm x Rm by assuming that the components of v , are listed last. We similarly split 1 to li and 1,. This decomposition yields a block structure for K .

Then (6.79) is equivalent to the following two relations:

Once the approximate solution uh is computed, we can obtain from the second relation that Ah,, = M - ' ( z , - KCiui- K,,u,). In the numerical examples below, we use uKn for finite element solutions on uniform meshes, and utd for finite element solutions on adaptive meshes. Since adaptive solutions are involved, numerical solution errors will be plotted against the number of degrees of freedom, rather than the meshsize.

E X A M P L E6 . 1 1 Let R = [O, 11 x [ O , 1 ] and F2 = r. The problem solved is: Find u E ~ ' ( 0 such ) that


Figure 6.1. Example 6.11, true solution

where

andfori = 1 , 2 ,

wi(x)=

{

(

1 - I ) ) if ri < 1, otherwise

(4 2 ( x z - x2,0) ( 2 ) 2 112 with ri = [ ( X I - x,,,) ] / ~ i In the experiments we let g = 1, xjt! = 0.8, x$t! = 0.1,

+

x(2) 2,0 -

= 0.3,

0.1, E I = 0 . 2 5 , ~ z= 0.2. We start with a coarse uniform triangulation shown on the left plot in Figure 6.2. Here, the interval 10, I ] is divided into l l h equal parts with h = 114 which is successively halved. The numerical solution corresponding to h = 11256 is taken as the "true" solution u, shown in Figure 6.1. We use the regular refinement technique (red-blue-green refinement), where the triangle is divided into four triangles by joining the midpoints of edges and adjacent triangles are refined in order to avoid hanging nodes. For a detailed description of this and other refinement techniques currently used see e.g. [I531 and references therein. Also, in order to improve the quality of triangulation, a

266


Figure 6.2. Example 6.11, initial partition and adaptively refined partition after 5 iterations

smoothing procedure is used after each refinement. For each triangle K of the triangulation we compute the triangle quality measure defined by

where hi, i = 1 , 2 , 3 , are the side lengths of the triangle K. Note that Q ( K ) = 1 if hl = h2 = h3. A triangle is viewed to be of acceptable quality if Q > 0.6, otherwise we modify the mesh by moving the interior nodes toward the center of mass of the polygon formed by the adjacent triangles. The adaptively refined triangulation after 5 iterations is shown on the right plot of Figure 6.2. To have an idea of the convergence behavior of the discrete Lagrange multipliers, we analyze the errors 11 X - Ah Il o , r corresponding to the sequence of uniform refinements. Here, X is the Lagrange multiplier corresponding to the parameter h = 11256. Figure 6.3 provides a comparison between the errors (ju - uKn11 l,n and h1I2( ( A - Ah jjO,r. The numerical convergence order of h1I211 X - Ah is obviously higher than that of Ilu - uXn 11 l , ~indicating , that the second term in the efficiency bounds (6.51) and (6.76) is expected to be of Graphs of Xh,1 and higher order compared to the first term Jju - uh with h = 11256 are provided in Figures 6.4 and 6.5. We use an adaptive procedure based on both residual type and gradient recovery type estimates to obtain a sequence of approximate solutions uzd. The adaptive finite element mesh after 5 adaptive iterations is shown on the right plot in Figure 6.2. The value ( / u ( (is~about , ~ 1.3228. Figures 6.6 and 6.7 contain the error values 11 u - uXn 11 1,a and I / u - uidI 11,a. We observe a substantial improvement of the efficiency using adaptively refined meshes. Figures 6.8 7):,K)1/2,I E { R ,G), where ~ I , K and 6.9 provide the values of 71 = are computed using either residual type estimator (I = R) or gradient recov-

(C,


1o2 1o3 Number of degrees of freedom

Figure 6.3. Example 6.11, 1

1o4

1 -~ u;fnlli,n (0)VS. h 1 ' 2 / I ~- Xh/10,r(A)

Figure 6.4. Example 6.11, plots of

A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY Side: ( l ) x [ O , l ]

Side [ O , l h ( O )

0 -0 5 -1 -1.5

0

0.2

0.4

06

0.8

1

-1.5

0

0.2

04

0.6

08

Y

x

0.3 Side, (O)x[O,l]

Figure 6.5. Example 6.11, plots of

Table 6.2. Example 6.11, numerical values of constants

ery type estimator (I = G) on both uniform and adapted meshes. Table 6.2 contains the values of CI computed for uniform and adaptive solutions:

It is seen from Table 6.2 that for this numerical example, the gradient recovery type error estimator provides a better prediction of the true error than the residual type error estimator, a phenomenon observed in numerous references. In general, we use the a posteriori error estimates only for the purpose of designing adaptive meshes, due to the presence of the unknown constants CR and CG


1o

-~ 10'

1o3 Number of degrees of freedom

1o2

I O*

Figure 6.6. Example 6.11, results based on residual type estimator, Ilu - u;tnlli,n ( 0 ) vs. 11u - uahdIl1,n(A)

1o

-~ 10'

I02 1o3 Number of degrees of freedom

I

10'

Figure 6.7. Example 6.11, results based on gradient recovery type estimator, Ilu - ~:"lll,n (0)vs. llu - u"hl11,n(4


4adapted mesh error adapted mesh I),

J 10'

Io2 1o3 Number of degrees of freedom

1o4

Figure 6.8. Example 6.11, results based on residual type estimator, 1)u- ~ ; l ~ and J l V~R ,on~ ~ V R on adapted mesh (A) uniform mesh (o) vs. ilu - u i d11 1 , and

:

10'

4adapted mesh error

1

1O-Z 10'

1oZ 10' Number of degrees of freedom

1 o4

Figure 6.9. Example 6.11, results based on gradient recovery type estimator, ilu and VG on uniform mesh ( 0 ) vs. Ilu - u P I I ~and , ~76 on adapted mesh (A)

~ x ~ l l ~ , ~

Error Analysis for Variational Inequalities of the Second Kind 10' adapted mesh (residual) 4- adapted mesh (gradient recovery) 4

10-21 10'

102 10' Number of degrees of freedom

1o4

Figure 6.10. Example 6.1 1, performance comparison between the two error estimators

For comparison of the performance between the two error estimators, we show in Figure 6.10 the errors of the adaptive solutions corresponding to the two error estimators. We observe that the two error estimators lead to very similar solution accuracy for same amount of degrees of freedom. More numerical examples can be found in [26].

6.6.

APPLICATION TO A FRICTIONAL CONTACT PROBLEM

In this section, we take the frictional contact problem discussed in Example 1.27 (page 32) to show that similar a posteriori error estimates can be derived for more complicated variational inequalities. Recall that R c IRd ( d 5 3 in applications) a Lipschitz domain. The unit outward normal vector v exists a.e. on the boundary r . We will use the product space ~ ' ( 0 = ()H I equipped with the norm d I : , n . Similarly, for w C R or w C I?, we use the space I V Il:,n = ~ ~ (=w( L) ~ ( w equipped ))~ with the norm lvl$,, = /vil;,,. Recall that the material is linearly elastic the fourth-order elasticity tensor @ : R x Sd -+ S d is assumed to be bounded, symmetric and positive definite in - - R. The boundary I? is partitioned as I? = I?D U F N U I?c with r D ,rNand rc relatively open and mutually disjoint, and meas(rD) > 0. The body is clamped on I'D; on the boundary part rNsurface tractions of density f E ( L ~ ( I ? N ) ) ~

27 2


are applied and on rc the body is in bilateral contact with a rigid foundation. The contact is frictional and is modeled by Tresca's law. Volume forces of density f E ( L ~ ( R act ) ) in ~ R. The function space for the displacement variable is V = {V E H ' ( R ) : vlr, = 0, vvlrc = O), with the inner product and norm

Over the space V , we use the following functionals:

a ( u , v )=

S,

C E ( U .) ~ ( vdx, )

Then the variational inequality of the frictional contact problem is to find u E V such that

This problem has a unique solution u in V , which is the minimizer of the energy functional 1 over the space V . In the analysis of a posteriori error estimators later, we will need the following characterization of the solution u of (6.80) through the use of a Lagrange multiplier: The unique solution u E V of the problem (6.80) is characterized by the existence of A, E (L" ( I ' c ) ) dsuch that

Turn now to a finite element approximation of the problem. Assume R has a polyhedral boundary I?. Let { P h }be a family of partitions of the domain R into straight-sided elements, and let V h C V be the associated standard finite element spaces of continuous piecewise polynomials of certain degree.


273

Corresponding to the partition P h , we use the symbols h K ,h,, E ( K ) , Eh, Eh,O as introduced in Section 6.1. In addition, we use &h,rN,EhJc with obvious meanings. The discrete formulation of the variational inequality (6.80) reads: Find u h E V h such that Like the continuous variational inequality (6.80), the discrete problem (6.83) has a unique solution. Analysis of the finite element approximation of such problems in the general context of variational inequalities is extensively discussed in [65], [66]. In the context of finite element approximations of a problem more general than the one considered in this paper, one can find in [81, Section 8.21 an optimal order a priori error estimate under additional solution regularity, and a convergence result without any additional solution regularity assumption. For derivation of a posteriori error estimates, we need the following characterization of the discrete solution through a Lagrange multiplier: The unique solution uh E V h of the problem (6.83) is characterized by the existence of ~ that ) ) ~ Xhr E ( ~ ~ ( rsuch

where vh, and u h , denote the tangential components of vh and uh, respectively. Then similar to the result in Section 6.3, we have the residual-type error estimator: Ilehll", c'&, where

r K and Ry are the interior and side residuals, respectively, defined by ( a h =

c.5( ~ h ) ) :


Figure 6.1I .

Example 6.12, physical setting

Moreover,

with discontinuous piecewise polynomial approximations f l , K , Xhr,r of residuals f l, Ah7, respectively. The results in Section 6.4 on gradient recovery-based error estimators can be similarly extended to the finite element solution of the frictional contact problem (6.80). Now we present numerical results on two two-dimensional problems. In all the examples no body forces are assumed to act and the body is in bilateral frictional contact with a rigid foundation on the part rc. The friction is modeled by the Tresca law with a given slip bound g. We use the adaptive algorithm stated at the beginning of Section 6.5, with p = 1.

EXAMPLE 6.12 We consider the physical setting shown in Figure 6.11. The domain R = (0,4) x (0,4) is the cross section of a three-dimensional linearly = (4) x elastic body and plane stress condition is assumed. On the part (0,4) the body is clamped. Oblique tractions act on the part (0) x (0,4) and the part (0,4) x (4) is traction free. Thus rrv= ((0) x (0,4))U ((0,4) x (4)). The contact part of the boundary is Fc = (0,4) x (0). The elasticity tensor C satisfies

rD


Figure 6.12. Example 6.12, initial mesh with 128 elements, 81 nodes

where E is the Young's modulus, u is the Poisson's ratio of the material and bij is the Kronecker symbol. We use the following data (the unit daN/rnm2 stands for "decanewtons per square milimeter"):

We start with the initial triangulation Pl shown in Figure 6.12. Here the interval [O, 41 is divided into 4 / h equal parts with h = 112 which is successively halved. The numerical solution corresponding to h = 1/64 is taken as the "exact" solution u . To have an idea of the convergence behavior of the discrete Lagrange multipliers, we compute the errors 1 A, - Ah, corresponding to the sequence of uniform refinements. Here, A, is the Lagrange multiplier corresponding to the parameter h = 1/64. Figure 6.13 provides a comparison of the errors )/ u - uxn11 and h1I2)I X I - Xhr I ojrc. The numerical convergence order of h1I2/Ix, - Ahrllo,rc is obviously higher than that of Ilu - uinIlv, indicating that the second term in the efficiency bound is expected to be of higher order compared to the first term llu - u h II$. Note that in the two-dimensional case, A, and Ah, are scalar valued functions, to be also denoted by X and Ah. The (approximate) calculation of Ah follows the procedure described in Section 6.5. Graphs of X h , ~and with h = 1/64 are provided in Figures 6.14 and 6.15. We use an adaptive procedure based on both residual type and gradient recovery type estimates to obtain a sequence of approximate solutions u i d . The deformed configuration and the adaptive finite element mesh after 4 adaptive iterations are shown in Figures 6.16 (residual type estimator) and 6.17 (gradient recovery type estimator). Figures 6.18 and 6.19 contain the error values / I u - u$IJ and ) / u- uidlJ v . We observe a substantial improvement of the


1o

-~ 1o2


Figure 6.14. Example 6.12, plot of

10'

Xh,1

efficiency using adaptively refined meshes. Figures 6.20 and 6.21 provide the values of 71 = ( C , 1):,,)1/2, I E { R , G I , where 7 1 , are ~ computed using either residual type estimator ( I = R) or a gradient recovery type estimator H ( I = G ) on both uniform and adapted meshes.

EXAMPLE 6.13 Consider the physical setting shown in Figure 6.22. The domain R = ( 0 , l O ) x ( 0 , 2 ) is the cross section of a three-dimensional linearly elastic body with plane stress condition assumed. On the part rD= ( 0 , l O ) x


421

0

05

I

15

2

25

3

Figure 6.15. Example 6.12, plot of 45

1

35

4

Xh,2

I

Figure 6.16. Example 6.12, results based on residual type estimator, deformed configuration 5583 elements. 2921 nodes

(2) the body is clamped. Horizontal tractions act on the part (0) x (0,2) and oblique tractions on (10) x (0,2). Here rc = (0,lO) x (0). The following data are used:


Figure 6.17. Example 6.12, results based on gradient recovery type estimator, deformed configuration with 5437 elements, 2832 nodes

J

1o2

Io3 Number of degrees of freedom

1o4

Figure 6.18. Example 6.12, results based on residual type estimator, Iu - u;tnllv(o) vs. llu - uPIIv (A)

As before we start with a coarse uniform triangulation Ph consisting of 160 triangular elements and 105 nodes depicted in Figure 6.23. The interval [O, 101 is divided into 1 0 / h equal parts and interval [O,21 is divided into 4 / h parts with h = 112. The numerical solution corresponding to h = 1/64 is taken as the


1o0 -A- adapted mesh


Figure 6.19. Example 6.12, results based on gradient recovery type estimator, Ilu - uinliv ( 0 ) VS. IIu - uahdllv (A)

Number of degrees of freedom

Figure 6.20. Example 6.12, results based on residual type estimator, Ilu - uXn uniform mesh ( 0 ) vs. llu - uidI / and 1 7 on ~ adapted mesh (A)

and 1 7 on ~


4adapted mesh error adapted mesh q,

1o2


1o4

Figure 6.21. Example 6.12, results based on gradient recovery type estimator, llu - u;lnllV and 1 7 on ~ uniform mesh (o) vs. ilu - U? dlV and 176. on adapted mesh (A)

-1

rigid obstacle

Figure 6.22. Example 6.13, physical setting

"exact" solution u.Figures 6.24 (residual type) and 6.25 (gradient recovery type) show the approximate solution and refined mesh after 4 consecutive refinements. Graphs of Xh,l and with h = 1/64 are provided in Figures l u - uKnllv, 6.26 and 6.27. Again, we compute the errors I h 1 1 2 ) I~T X h 7 ) ) ~ ; rand C 711, I E {R, G), whose values are provided in Figures 6.28-6.32.

Iu

~ ~ ~ l l ~


28 1

w Figure 6.23. Example 6.13, initial mesh with 160 elements, 105 nodes

Fkure 6.24. Example 6.13, results based on residual type estimator, deformed configuration with 5222 elements, 2755 nodes

Figure 6.25. Example 6.13, results based on gradient recovery type estimator, deformed configuration with 4964 elements, 2624 nodes


Figure 6.26. Example 6.13, plot of Ah,]

Figure 6.27. Example 6.13, plot of

Ah,2


1o0

+ uniform mesh 4adapted mesh

10-21 1

o2

I o3

1o4


Figure 6.28. Example 6.13, results based on residual type estimator, Ilu - u;tnilv ( 0 ) vs. llu - uahdIIv (A)

+ uniform mesh 4- adapted mesh

10-21 1o2

1o3

1o4


Figure 6.29. Example 6.13, results based on gradient recovery type estimator, Ilu - u;tnllv ( 0 ) VS. Ilu - uahdIIv (A)


+ solution -A- lambda

- ~L

1o

1o2


1o4


10'

10.'

adapted mesh error

1o

-~ 102

Figure 6.31. Example 6.13, results based on residual type estimator, Jlu- uznllVand 1 7 on ~ uniform mesh ( 0 ) vs. 1 lu - uRd/ I and 1 7 on ~ adapted mesh (A)

I

1

S9.0

% qsau umjun mia qsau uiojlun -s

Ei===

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Index

C(Q, 7

cm(a),7 C? (n),7 C m ( 2 ) ,7

cO,a (Q,8 cm>"(n), 8 H 1 ( R ) ,13 H ! ~(n),16 H"(R, 10 inner product, 10 norm, 10 H t ( R ) >11 H-'(a),13 H ' / ~ ( ~ R12) , L m ( O ) ,8 L p ( R ) 8, L f O C ( R8) , w k , p ( R ) 9, norm, 9 seminorm, 15 w $ ' P ( 11 ~), Rd,5 inner product, 5 norm, 5 sd,5 inner product, 5 norm, 5 a posteriori error estimate, 3 a priori error estimate, 44 auxiliary function, 66 admissible, 73 best constant, 19 bilinear form, 18 bounded, 18 continuous, 18 elliptic, 18 symmetric, 18

CCa's inequality, 37 Carathtodory function, 58 Cauchy-Schwarz inequality, 6 ClCment-type interpolation, 240 coercive function, 56 compact embedding, 14 conjugate functional, 57 convergence, 6 strong, 6 weak, 6 convex function, 48 characterization, 56 continuity, 52 strictly, 48 strongly, 63 convex set, 47 density theorems, 11 directional derivative, 55 domain, 5 Lipschitz, 10 dual problem, 57 dual space, 5 dual variable, 66 admissible, 73 dual-weighted residual technique, 4 duality pairing, 5 duality theory, 59 effective domain, 48 elliptic variational inequality first kind, 30 second kind, 30 embedding, 13 energy difference, 61 epigraph, 49 equivalent norms, 14 error estimator, 235 gradient recovery-based, 255

A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY residual-based, 248, 273 error indicator. 235 finite element interpolant, 43 finite element method, 38 h-p-version, 38 h-version, 38 p-version, 38 finite element space, 41 finite elements affine-equivalent, 41 frictional contact problem, 32 Gdteaux derivative, 55 Galerkin method, 36 gradient recovery-based error estimate, 255 Holder continuity, 8 Hahn-Banach theorem, 50 heat conduction problem, 119 coefficient idealization, 121 insulation boundary condition, 122 linearization, 160 temperature boundary condition, 124 Hencky material, 144 idealizations in linear problems, 67 boundary condition, 100 coefficient, 68 domain, 106 right-hand side, 91 indicator function, 49,53 KaEanov method, 204 convergence, 205 elastoplasticity, 226 quasi-Newtonian flow problem, 219 stationary conservation law, 209 Kom inequality, 34 Lagrange multiplier, 238 Lax-Milgram Lemma, 18 linearization, 127 Bingham flow, 176 heat conduction problem, 160 nonlinear elasticity, 143 obstacle problem, 182 problem with small parameter, 169 quasilinear problem, 173 Lipschitz continuity, 7 Lipschitz domain, 10 lower semicontinuous (l.s.c.),49 mathematical model, 1 basic, 1 idealized, 1

8 meas (a), mesh, 39 mesh parameter, 40 multi-index notation. 7 normal derivative, 13 obstacle problem, 3 1, 182, 193 partition, 39 regular, 40 PoincarC inequality, 19 primal problem, 57 proper functional, 29 reference element, 39 reflexive Banach space, 6 regularization method, 196 a posteriori error estimate, 200 convergence, 197 elastoplasticity, 226 obstacle problem, 193 residual-based error estimate, 248, 273 Riesz representation theorem, 6 Ritz method, 37 separation of convex sets, 50 singularity, 25 smoothness of boundary, 10 Sobolev space, 9 strain deviator, 144 stress deviator, 144 subdifferential, 53 subgradient, 53 sublinear functional, 50 summation convention, 5 support, 7 support functional, 53 torsion problem, 112 normalized Prandtl's stress function, 114 Prandtl's stress function, 113 warping function, 112 trace, 12 triangulation, 39 regular, 40 variational inequality, 29 weak derivative, 9 weak formulation, 16 weakly lower semicontinuous (w.l.s.c.), 49 Young inequality, 58 generalized, 58

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