Stability Estimates for Hybrid Coupled Domain Decomposition Methods
Domain decomposition methods are a well established tool for an efficient numerical solution of partial differential equations, in particular for the coupling of different model equations and of different discretization methods. Based on the approxi
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Olaf Steinbach
Stability Estimates for Hybrid Coupled Domain Decomposition Methods
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Author Olaf Steinbach Institut f¨ur Angewandte Analysis und Numerische Simulation Universit¨at Stuttgart Pfaffenwaldring 57 70569 Stuttgart, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 65N30, 65N38, 65N55, 35J25 ISSN 0075-8434 ISBN 3-540-00277-4 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, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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 c Springer-Verlag Berlin Heidelberg 2003 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 specif ic 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: 10825795
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Saddle Point Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Finite Element Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Quasi Interpolation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 7 10 14 18 22
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Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Piecewise Linear Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dual Finite Element Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Higher Order Finite E
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