Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities

488 Pages·1998·11.219 MB·English

by  Zi Cai Li (auth.)

Combined Methods for Elliptic Equations with
Singularities, Interfaces and Infinities
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 444
Combined Methods for Elliptic
Equations with Singularities,
Interfaces and Infinities
by
Zi eai Li
Department ofA pplied Mathematics,
National Sun Yat-sen University,
Kaohsiung, Taiwan
KLUWER ACADEMIC PUBLISHERS
DORDRECHT/BOSTON/LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13: 978-1-4613-3340-1 e-ISBN-13: 978-1-4613-3338-8
DOl: 978-1-4613-3338-8
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
Sold and distributed in North, Central and South America
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In all other countries, sold and distributed
by Kluwer Academic Publishers,
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Printed on acid-free paper
All Rights Reserved
© 1998 Kluwer Academic Publishers
Softcover reprint of the hardcover 18t edition 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
To my friends)
John Fraser and Elizabeth MacCallum.
CONTENTS
PREFACE xv
ACKNOWLEDGEMENTS xix
INTRODUCTION
XXI
Part I SINGULARITIES, TREATMENTS AND
COMBINATIONS 1
1 DIFFERENT NUMERICAL METHODS 5
1.1 Descriptions of Elliptic Equations 5
1.2 Finite Element Method 7
1.3 Ritz-Galerkin Method 11
1.4 Boundary Approximation Method 13
1.5 Finite Difference Method 16
1.5.1 Interior Equations 17
1.5.2 Boundary Difference Equations 18
1.5.3 View of FDM as FEM 20
1.6 Finite Volume Method 24
1.6.1 Conservative Law 24
1.6.2 Partition without Obtuse Triangles 26
1.6.3 Interface Conditions 28
1.6.4 View of FVM as FDM 30
1.6.5 Partition with Delaunay Triangulation 32
1.6.6 New View of FVM as Galerkin FEM 33
VB
Vlll COMBINATIONS FOR SINGULARITY PROBLEMS
1. 7 Boundary Element Method 35
1. 7.1 Preliminary Lemmas and Theorems 35
1. 7.2 Discrete Approximation 37
1. 7.3 Galerkin Approach 38
1.7.4 Natural BEM 39
1.8 Collocation Method 40
1.8.1 Algorithms 40
1.8.2 Viewpoint of BAM 42
1.8.3 Viewpoint of FEM 44
1.9 Least Squares Method 45
1.10 Comparisons 47
2 SINGULARITIES AND TREATMENTS 49
2.1 Singularity of Laplace Equation on Polygons 52
2.1.1 Neumann-Dirichlet Conditions 52
2.1.2 Analysis on Singularity 57
2.1.3 Other Particular Solutions 58
2.1.4 Motz Problem 60
2.1.5 Reduced Convergence Rates Caused by Singularities 61
2.1.6 Particular Solutions near Infinity 62
2.2 Conformal Transformation Method 63
2.2.1 Basic Methods 63
2.2.2 Algorithms for Expansion Solutions 69
2.2.3 First Ten Leading Coefficients 72
2.3 Local Refinements 72
2.4 Singular Elements 77
2.4.1 Modifying Shape Functions 79
2.4.2 Modifying Reference Nodes 79
2.5 Infinity Element Method 80
2.6 Singular Function Methods 81
Contents IX
2.7 Combinations of h- and p-Versions 83
2.8 Combinations of FEM and BEM 84
2.8.1 Combination for Angular Singularity 85
2.8.2 Combination for Unbounded Domain Problems 87
2.8.3 Comparisons with Combined Methods of RGM and FEM 89
2.9 Combined Methods 91
2.9.1 Introduction 91
2.9.2 General Description 92
2.9.3 Aspects in Analysis 94
Part II COMBINED METHODS 99
3 BOUNDARY APPROXIMATION METHODS 103
3.1 Notations and Preliminaries 104
3.2 Approximation Problems 106
3.3 Error Estimates 109
3.4 Debye-Huckel Equation 113
3.5 Stability Analysis 119
3.6 Two Models of Singularity Problems 125
3.6.1 Motz Problem 126
3.6.2 Comparisons with CTM 131
3.6.3 Crack-Infinity Problem 133
4 COMBINATIONS OF RGM AND FEM 139
4.1 Nonconforming Approach for Laplace Equation 140
4.2 Error Estimates 146
4.3 Coupling Strategy 152
4.4 Stability of Numerical Solutions 155
4.5 Elliptic Equations 157
4.6 Choice of Polynomial Bases 164
x COMBINATIONS FOR SINGULARITY PROBLEMS
4.7 Uniform Vh - Ellipticity and Stability 166
4.8 Numerical Experiments for Motz Problem 169
5 COMBINATIONS OF VARIOUS FEMS 175
5.1 Description of Methods 177
5.2 Programming Technique 181
5.3 Error Bounds and Coupling Strategy 183
5.4 Combinations of M(2 1)- and K(> M)-Order Lagrange FEMs 186
5.5 Proof of Theorem 5.1 187
5.6 Reduced Rate of Convergence 193
5.7 Numerical Experiments of RRC Coupling 200
Part III COUPLING TECHNIQUES 205
6 LAGRANGE MULTIPLIERS AND OTHER
COUPLING TECHNIQUES 209
6.1 Introduction 209
6.2 Interpretation of Nonconforming Combinations as Lagrange
Multipliers 210
6.3 General Approaches of Lagrange Multipliers 212
6.4 Overview of Coupling Techniques without Extra Variables 220
7 PENALTY TECHNIQUES 225
7.1 Description of Coupling Techniques 226
7.2 Error Bounds in Presence of Variational Crimes 229
7.3 Coupling Strategy Between (L + 1) and h 235
7.4 Numerical Experiments 241
7.5 Relation to Nonconforming Combinations 243
8 SIMPLIFIED HYBRID METHODS 251
8.1 Description of Coupling 251