Finite Element Solution of Boundary Value Problems (eBook)

Theory and Computation

O. Axelsson, V. A. Barker (Autoren)

Werner Rheinboldt (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
452 Seiten
Elsevier Science (Verlag)
9781483260563 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Finite Element Solution of Boundary Value Problems: Theory and Computation 4
Copyright Page 5
Table of Contents 8
Dedication 6
Preface 10
Acknowledgments 14
List of Symbols 16
CHAPTER 1. Quadratic Functionals on Finite-Dimensional Vector Spaces 20
Introduction 20
1.1 Quadratic Functionals 21
1.2 The Method of Steepest Descent 28
1.3 The Conjugate Gradient Method 37
1.4 The Preconditioned Conjugate Gradient Method 47
Exercises 75
References 81
CHAPTER 2. Variational Formulation of Boundary Value Problems: Part I 83
Introduction 83
2.1 The Euler-Lagrange Equation for One-Dimensional Problems 84
2.2 Natural and Essential Boundary Conditions 92
2.3 Problems in Two and Three Dimensions 99
2.4 Boundary Value Problemsin Physics and Engineering 111
Exercises 115
References 119
CHAPTER 3. Variational Formulation of Boundary Value Problems: Part II 120
Introduction 120
3.1 The Concept of Completion 121
3.2 The Lax-Milgram Lemma and Applications 137
3.3 Regularity, Symbolic Functions, and Green's Functions 149
Exercises 158
References 163
CHAPTER 4. The Ritz–Galerkin Method 164
Introduction 164
4.1 The Ritz Method 165
4.2 Error Analysis of the Ritz Method 170
4.3 The Galerkin Method 172
4.4 Application of the Galerkin Method to Noncoercive Problems 174
Exercises 179
References 181
CHAPTER 5. The Finite Element Method 182
Introduction 182
5.1 Finite Element Basis Functions 184
5.2 Assembly of the Ritz–Galerkin System 199
5.3 Isoparametric Basis Functions 226
5.4 Error Analysis 233
5.5 Condition Numbers 251
5.6 Singularities 259
Exercises 266
References 284
CHAPTER 6. Direct Methods for Solving Finite Element Equations 287
Introduction 287
6.1 Band Matrices 288
6.2 Direct Methods 298
6.3 Special Techniques 308
6.4 Error Analysis 327
Exercises 336
References 344
CHAPTER 7. Iterative Solution of Finite Element Equations 346
Introduction 346
7.1 SSOR Preconditioning 347
7.2 Preconditioning by Modified Incomplete Factorization : Part I 356
7.3 Preconditioning by Modified Incomplete Factorization : Part II 379
7.4 Calculation of Residuals: Computational Labor and Stability 387
7.5 Comparison of Iterative and Direct Methods 402
7.6 Multigrid Methods 411
Exercises 426
References 438
APPENDIX A: Chebyshev Polynomials 441
Index 446

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-13: 9781483260563 / 9781483260563