hp-Finite Element Methods for Singular Perturbations
Many partial differential equations arising in practice are parameter-dependent problems that are of singularly perturbed type. Prominent examples include plate and shell models for small thickness in solid mechanics, convection-diffusion problems in flui
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Jens M. Melenk
hp-Finite Element Methods for Singular Perturbations
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Author Jens M. Melenk Max Planck Institute for Mathematics in the Sciences Inselstr. 22 04103 Leipzig Germany e-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Melenk, Jens M.: hp-finite element methods for singular perturbations / Jens M. Melenk. Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; Vol. 1796) ISBN 3-540-44201-4
Mathematics Subject Classification (2000): 65N30, 65N35, 58J37, 35J25 ISSN 0075-8434 ISBN 3-540-44201-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10891013
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Preface
Many partial differential equations arising in practice are parameter-dependent problems and are of singularly perturbed type for small values of this parameter. These include various plate and shell models for small thickness in solid mechanics, the convection-diffusion equation, the Oseen equation, and NavierStokes equations in fluid flow problems where the fluid is assumed to have small viscosity, and finally equations arising in semi-conductor device modelling. Analysis of such equations by numerical methods such as the finite element method is an important task in today’s computational practice. A significant design aspect of numerical methods for such parameter-dependent problems is robustness, that is, that the performance of the numerical method is independent of, or at least fairly insensitive to, the parameter. Numerous methods have been proposed and analyzed both theoretically and computationally for a variety of singularly perturbed problems—we merely refer at this point to the three recent monographs [97, 99, 108] and their extensive bibliographies. Most numerical methods employed in the study
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