Galerkin Finite Element Methods for Parabolic Problems
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1054
Vidar Thomee
Galerkin Finite Element Methods for Parabolic Problems
Spri nger-\tenaq Berlin Heidelberg New York Tokyo 1984
Author
Vidar Thornee Department of Mathematics Chalmers University of Technology and the University of G6teborg S -41296 - G6teborg, Sweden
AMS Subject Classifications (1980): 65N30 ISBN 3-540-12911-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12911-1 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
The purpose of this work is to present, in an essentially self-contained form, a survey of the mathematics of Galerkin finite element methods as applied to parabolic problems. The selection of topics is not meant to be exhaustive, but rather reflects the author's involvement in the field over the past ten years. The goal has been mainly pedagogical, with emphasis on collecting ideas and methods of analysis in simple model situations, rather than on pursuing each approach to its limits. The notes thus summarize recent developments, and the reader is often referred to the literature for more complete results on a given topic. Because the formulation and analysis of Galerkin methods for parabolic problems are generally based on facts concerning the corresponding stationary elliptic problems, the necessary elliptic results are included in the text for completeness. The following is an outline of the contents of the notes: In the introductory Chapter 1 we consider the simplest Galerkin finite element method for the standard initial boundary value problem with homogeneous Dirichlet boundary conditions on a bounded domain for the heat equation, using the standard associated weak formulation of the problem and employing first piecewise linear and then more general piecewise polynomial approximating functions vanishing on the boundary of the domain. For this model problem we demonstrate the basic error estimates in energy and mean square norms, first for the semidiscrete problem resulting from discretization in the space variables only and then also for the most commonly used completely discrete schemes obtained by discretizing the semidiscrete equation with respect to the time variable. In the following five chapters we consider several extensions and generalizations of these results in the case of the semidiscrete approximation, and show error estimates in a variety of norms. First, in Chapter 2, we express the semidiscrete problem by means of an approximate solution operator for the elliptic problem in a
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