Table Of ContentDMVSeminar
Band 15
Springer Science+Business Media, LLC
Albert H. Schatz
Vidar Thomee
Wolfgang L.Wendland
Mathematical Theory of
Finite and Boundary
Element Methods
Authors:
Albert H. Schatz Vidar Thomee
Dept. of Mathematics Dept. of Mathematics
Cornell University Chalmers University of Technology
Ithaca, N.Y. 14853 University of Goteborg
USA S-41296 Goteborg
Sweden
Wolfgang L. Wendland
University of Stuttgart
Mathematisches Institut A
Pfaffenwaldring 57
D-7000 Stuttgart 80
Federal Republic of Germany
The seminar was made possible through the support of the
Stiftung Volkswagenwerk
Deutsche Bibliothek Cataloguing-in-Publication Data
MathematicaI theory of finite and boundary element methods /
A. H. Schatz ; V. Thomee ; W. L. Wendland. - Basel ; Boston
Berlin : Birkhiiuser, 1990
(DMV-Seminar ; Bd. 15)
ISBN 978-3-7643-2211-3 ISBN 978-3-0348-7630-8 (eBook)
DOI 10.1007/978-3-0348-7630-8
NE: Schatz, Albert H. [Mitverf.]; Thomee, Vidar [Mitverf.]; Wendland,
Wolfgang L. [Mitverf.]; Deutsche Mathematiker-Vereinigung:
DMV-Seminar
This work is subject to copyright. AII 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 >Nerwertungsgesellschaft Wort«,
Munich.
© 1990 Springer Science+Business Media New York
Originally published by Birkhiiuser Verlag Base! in 1990
ISBN 978-3-7643-2211-3
5
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7
Part I: An Analysis of the Finite Element Method for
Second Order Elliptic Boundary Value Problems
by A. H. Schatz
O. Introduction........................................ 9
1. Some function spaces, notation and preliminaries ................ 11
2. Some finite element spaces and their properties ................. 21
3. Orthogonal projections onto finite element spaces in L2, in Hl and H5 ... 39
4. Galerkin finite element method for second order elliptic boundary value
problems. Basic Hl and L2 estimates . . . . . . . . . . . . . . . . . . . . . . . .. 51
5. Indefinite second order elliptic problems . . . . . . . . . . . . . . . . . . . . .. 67
6. Local error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
7. An introduction to grid refinement. An application to boundary value
problems with non-convex corners .... . . . . . . . . . . . . . . . . . . . . .. 85
8. Maximum norm estimates for the L2 projection. A method using weighted
norms ............................................ 99
9. Maximum norm estimates for the Galerkin finite element method for
second order elliptic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References ......................................... 126
Part II: The Finite Element Method for Parabolic Problems
by Vidar Thomee
1. Introduction........................................ 135
2. Non-smooth data error estimates for the semidiscrete problem . . . . . . .. 161
3. Completely discrete schemes .............................. 173
4. A nonlinear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Part III: Boundary Element Methods for Elliptic Problems
by W. L. Wendland
1 Boundary Integral Equations
1.1 The exterior Neumann problem for the Laplacian . . . . . . . . . . . . . 223
1.2 Exterior viscous flow problems . . . . . . . . . . . . . . . . . . . . . . . . . 230
1.3 Scattering problems in acoustics ........................ 232
1.4 Some problems of elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . 234
1.5 The boundary integral equations of the direct approach for general
elliptic boundary value problems of even order . . . . . . . . . . . . . . . 237
6
2 The Characterization of Boundary Integral Operators and Galerkin
Boundary Element Methods
2.1 The representation and the order of boundary integral operators 239
2.2 Variational formulation and strong ellipticity . . . . . . . . . . . . . . . . 244
2.3 Boundary element Galerkin methods ..................... 247
3 CoUocation Methods
3.1 Collocation with smoothest splines of piecewise odd polynomials 257
3.2 Naive spline collocation for n = 2 on almost uniform partitions .... 264
4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Preface
These are the lecture notes of the seminar "Mathematische Theorie der finiten Element
und Randelementmethoden" organized by the "Deutsche Mathematiker-Vereinigung"
and held in Dusseldorf from 07. - 14. of June 1987.
Finite element methods and the closely related boundary element methods nowadays
belong to the standard routines for the computation of solutions to boundary and initial
boundary value problems of partial differential equations with many applications as e.g.
in elasticity and thermoelasticity, fluid mechanics, acoustics, electromagnetics, scatter
ing and diffusion. These methods also stimulated the development of corresponding
mathematical numerical analysis. I was very happy that A. Schatz and V. Thomee
generously joined the adventure of the seminar and not only gave stimulating lectures
but also spent so much time for personal discussion with all the participants.
The seminar as well as these notes consist of three parts:
1. An Analysis of the Finite Element Method for Second Order Elliptic Boundary
Value Problems by A. H. Schatz.
II. On Finite Elements for Parabolic Problems by V. Thomee.
III. I30undary Element Methods for Elliptic Problems by \V. L. Wendland.
The prerequisites for reading this book are basic knowledge in partial differential
equations (including pseudo-differential operators) and in numerical analysis. It was
not our intention to present a comprehensive account of the research in this field, but
rather to give an introduction and overview to the three different topics which shed
some light on recent research.
These lecture notes could have been ready two years ago if I were not the bad slow
guy. The more grateful I am to A. Schatz, V. Thomee and the I3irkhiiuser Verlag for
their patience: I am happy that the notes now are available. I want to thank the DMV
8
and in particular my colleague G. Fischer for encouraging and organizing the seminar;
the "Volkswagen-Stiftung" for their generous support; and the hosts of 'SchloB Mickeln'
for their great efforts in making our stay a most pleasant one. I am espescially indebted
to A. Schatz and V. Thomee for their wonderful contributions to the seminar and to all
participants for their interest, the lively discussions and their good company.
Stuttgart, Mai 1990. Wolfgang Wendland
9
An Analysis of the Finite Element Method for Second Order
Elliptic Boundary Value Problems. An Introduction
Alfred H. Schatz
Cornell University
Ithaca. N.Y. U.S.A.
O. Introduction, Table of Contents.
These notes represent an expansion of lectures given
at Dusseldorf in the summer of 1987. It is hoped that they will
provide an introduction to the mathematical analysis of the
finite element method for second order elliptic boundary value
problems. They are geared towards students who have a good
background in analysis but are not familiar with the finite
element method. Thus we start at the beginning with the aim of
understanding the method and developing basic techniques and
ideas which seem to be fundamental to it. The treatment here is
neither exhaustive nor general. Rather. topics are covered
which we feel address important problems whose solutions make
use of basic techniques. We have attempted to give self-
contained proofs wherever possible. It is hoped that these will
be adequate in order that the student be able to understand some
recent results in the field. For the convenience of the student
a bibliography has been included which covers some topics not
treated in these notes.
