Numerical Methods for Nonlinear Partial Differential Equations
The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for lin
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Sören Bartels
Numerical Methods for Nonlinear Partial Differential Equations
Springer Series in Computational Mathematics Volume 47
Editorial Board R. Bank R.L. Graham W. Hackbusch J. Stoer R. Varga H. Yserentant
More information about this series at http://www.springer.com/series/797
Sören Bartels
Numerical Methods for Nonlinear Partial Differential Equations
123
Sören Bartels Abteilung für Angewandte Mathematik Albert-Ludwigs-Universität Freiburg Freiburg Germany
ISSN 0179-3632 Springer Series in Computational Mathematics ISBN 978-3-319-13796-4 ISBN 978-3-319-13797-1 DOI 10.1007/978-3-319-13797-1
(eBook)
Library of Congress Control Number: 2014957868 Mathematics Subject Classification (2010): 65N30, 65N12, 65M12, 65M22, 65K15, 49M15, 49M29 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Differential Equations and Numerical Methods . . . . . . . . . . 1.2 Guidelines for the Development of Approximation Schemes. 1.3 Analytical and Numerical Foundations. . . . . . . . . . . . . . . . 1.4 Approximation of Classical Formulations . . . . . . . . . . . . . . 1.5 Numerical Methods for Extended Formulations. . . . . . . . . . 1.6 Objectives and Acknowledgments . . . . . . . . . . . . . . . . . . .
Part I 2
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11 11 11 11 12 13 14 14 15 15 16 17 17 18 19 21 22 26
Analytical and Numerical Foundations
Analytical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Variational Model Problems . . . . . . . . . . . . . . . . . . . . . 2.1.1 Deflection of a Membrane . . . . . . . . . . . . . . .
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