A Posteriori Error Analysis via Duality Theory With Applications in
This volume 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. The auth
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Advances in Mechanics and Mathematics Volume 8
Series Editors: David Y. Gao Virginia Polytechnic Institute and State University, U.S.A.
Ray W. Ogden University of Glasgow, U.K.
Advisory Editors:
I. Ekeland University of British Columbia, Canada K.R. Rajagopal Texas A M University, U.S.A.
T . Ratiu Ecole Polytechnique, Switzerland
W. Yang Tsinghua University, P.R. China
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.
Library of Congress Cataloging-in-Publication Data
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
SPIN 11336112
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
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A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY
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
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