Axelsson / Barker / Rheinboldt Finite Element Solution of Boundary Value Problems

Theory and Computation
1. Auflage 2014
ISBN: 978-1-4832-6056-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 452 Seiten, Web PDF

Finite Element Solution of Boundary Value Problems: Theory and Computation provides an introduction to both the theoretical and computational aspects of the finite element method for solving boundary value problems for partial differential equations. This book is composed of seven chapters and begins with surveys of the two kinds of preconditioning techniques, one based on the symmetric successive overrelaxation iterative method for solving a system of equations and a form of incomplete factorization. The subsequent chapters deal with the concepts from functional analysis of boundary value problems. These topics are followed by discussions of the Ritz method, which minimizes the quadratic functional associated with a given boundary value problem over some finite-dimensional subspace of the original space of functions. Other chapters are devoted to direct methods, including Gaussian elimination and related methods, for solving a system of linear algebraic equations. The final chapter continues the analysis of preconditioned conjugate gradient methods, concentrating on applications to finite element problems. This chapter also looks into the techniques for reducing rounding errors in the iterative solution of finite element equations. This book will be of value to advanced undergraduates and graduates in the areas of numerical analysis, mathematics, and computer science, as well as for theoretically inclined workers in engineering and the physical sciences.

Autoren/Hrsg.

Axelsson, O.

Barker, V. A.

Rheinboldt, Werner

Inhaltsverzeichnis

1;Front Cover;1
2;Finite Element Solution of Boundary Value Problems: Theory and Computation;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;10
7;Acknowledgments;14
8;List of Symbols;16
9;CHAPTER 1. Quadratic Functionals on Finite-Dimensional Vector Spaces;20
9.1;Introduction;20
9.2;1.1 Quadratic Functionals;21
9.3;1.2 The Method of Steepest Descent;28
9.4;1.3 The Conjugate Gradient Method;37
9.5;1.4 The Preconditioned Conjugate Gradient Method;47
9.6;Exercises;75
9.7;References;81
10;CHAPTER 2. Variational Formulation of Boundary Value Problems: Part I;83
10.1;Introduction;83
10.2;2.1 The Euler-Lagrange Equation for One-Dimensional Problems;84
10.3;2.2 Natural and Essential Boundary Conditions;92
10.4;2.3 Problems in Two and Three Dimensions;99
10.5;2.4 Boundary Value Problemsin Physics and Engineering;111
10.6;Exercises;115
10.7;References;119
11;CHAPTER 3. Variational Formulation of Boundary Value Problems: Part II;120
11.1;Introduction;120
11.2;3.1 The Concept of Completion;121
11.3;3.2 The Lax-Milgram Lemma and Applications;137
11.4;3.3 Regularity, Symbolic Functions, and Green's Functions;149
11.5;Exercises;158
11.6;References;163
12;CHAPTER 4. The Ritz–Galerkin Method;164
12.1;Introduction;164
12.2;4.1 The Ritz Method;165
12.3;4.2 Error Analysis of the Ritz Method;170
12.4;4.3 The Galerkin Method;172
12.5;4.4 Application of the Galerkin Method to Noncoercive Problems;174
12.6;Exercises;179
12.7;References;181
13;CHAPTER 5. The Finite Element Method;182
13.1;Introduction;182
13.2;5.1 Finite Element Basis Functions;184
13.3;5.2 Assembly of the Ritz–Galerkin System;199
13.4;5.3 Isoparametric Basis Functions;226
13.5;5.4 Error Analysis;233
13.6;5.5 Condition Numbers;251
13.7;5.6 Singularities;259
13.8;Exercises;266
13.9;References;284
14;CHAPTER 6. Direct Methods for Solving Finite Element Equations;287
14.1;Introduction;287
14.2;6.1 Band Matrices;288
14.3;6.2 Direct Methods;298
14.4;6.3 Special Techniques;308
14.5;6.4 Error Analysis;327
14.6;Exercises;336
14.7;References;344
15;CHAPTER 7. Iterative Solution of Finite Element Equations;346
15.1;Introduction;346
15.2;7.1 SSOR Preconditioning;347
15.3;7.2 Preconditioning by Modified Incomplete Factorization : Part I;356
15.4;7.3 Preconditioning by Modified Incomplete Factorization : Part II;379
15.5;7.4 Calculation of Residuals: Computational Labor and Stability;387
15.6;7.5 Comparison of Iterative and Direct Methods;402
15.7;7.6 Multigrid Methods;411
15.8;Exercises;426
15.9;References;438
16;APPENDIX A: Chebyshev Polynomials;441
17;Index;446