Superconvergence in Galerkin Finite Element Methods
This book is essentially a set of lecture notes from a graduate seminar given at Cornell in Spring 1994. It treats basic mathematical theory for superconvergence in the context of second order elliptic problems. It is aimed at graduate students and resear
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1605
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1605
Springer
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Lars B. Wahlbin
Superconvergence in Galerkin Finite Element Methods
Springer
Author Lars B. Wahlbin Department of Mathematics White Hall Cornell University Ithaca, NY 14853,USA E-mail: [email protected]
Library of Congress Cataloging-in-Publication Data. Wahlbin, Lars B., 1945Superconvergence in Galerkin finite element methods/Lars B. Wahlbin. p.cm. (Lecture notes in mathematics; 1605) Includes bibliographical references (p. -) and index. ISBN 3-540-60011-6 (acid-free) 1. Differential equation, Elliptic - Numerical solutions. 2. Convergence. 3. Galerkin methods. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1605. QA3. L28 no. 1605 [QA377] 510s-dc20 [515' .353]
Mathematics Subject Classification (1991): 65N30, 65N 15
ISBN 3-540-60011-6 Springer-Verlag Berlin Heidelberg New York
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46/3142-543210 - Printed on acid-free paper
Preface. These notes are from a graduate seminar at Cornell in Spring 1994. They are devoted mainly to basic concepts of superconvergence in second-order timeindependent elliptic problems. A brief chapter-by-chapter description is as follows: Chapter 1 considers onedimensional problems and is intended to get us moving quickly into the subject matter of superconvergence. (The results in Sections 1.8 and 1.10 are new.) Some standard results and techniques used there are then expounded on in Chapters 2 and 3. Chapter 4 gives a few selected results about superconvergence in L2-projections in any number of space dimensions. In Chapter 5 we elucidate local maximumnorm error estimates in second order elliptic partial differential equations and the techniques used in proving them, without aiming for complete detail. Theorems 5.5.1 and 5.5.2 are basic technical results; they will be used over and over again in the rest of the notes. In Chapters 6 through 12 we treat a variety of topics in superconvergence for second order elliptic problems. Some are old and established, some are very recent and not yet published. Some of the earlier contributions have benefitted from later sharpening of tools, in particular with respect to local maximum-norm estimates. In Chapter 6 we consider tensor-product elements. Using ideas of [Douglas, D
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