Finite Element Solution of Boundary Value Problems

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

Preface
Acknowledgments
List of Symbols

1. Quadratic Functionals on Finite-Dimensional Vector Spaces
   Introduction
   1.1 Quadratic Functionals
   1.2 The Method of Steepest Descent
   1.3 The Conjugate Gradient Method
   1.4 The Preconditioned Conjugate Gradient Method
   Exercises
   References
2. Variational Formulation of Boundary Value Problems: Part I
   Introduction
   2.1 The Euler-Lagrange Equation for One-Dimensional Problems
   2.2 Natural and Essential Boundary Conditions
   2.3 Problems in Two and Three Dimensions
   2.4 Boundary Value Problems in Physics and Engineering
   Exercises
   References
3. Variational Formulation of Boundary Value Problems: Part II
   Introduction
   3.1 The Concept of Completion
   3.2 The Lax-Milgram Lemma and Applications
   3.3 Regularity, Symbolic Functions, and Green's Functions
   Exercises
   References
4. The Ritz-Galerkin Method
   Introduction
   4.1 The Ritz Method
   4.2 Error Analysis of the Ritz Method
   4.3 The Galerkin Method
   4.4 Application of the Galerkin Method to Noncoercive Problems
   Exercises
   References
5. The Finite Element Method
   Introduction
   5.1 Finite Element Basis Functions
   5.2 Assembly of the Ritz-Galerkin System
   5.3 Isoparametric Basis Functions
   5.4 Error Analysis
   5.5 Condition Numbers
   5.6 Singularities
   Exercises
   References
6. Direct Methods for Solving Finite Element Equations
   Introduction
   6.1 Band Matrices
   6.2 Direct Methods
   6.3 Special Techniques
   6.4 Error Analysis
   Exercises
   References
7. Iterative Solution of Finite Element Equations
   Introduction
   7.1 SSOR Preconditioning
   7.2 Preconditioning by Modified Incomplete Factorization: Part I
   7.3 Preconditioning by Modified Incomplete Factorization: Part II
   7.4 Calculation of Residuals: Computational Labor and Stability
   7.5 Comparison of Iterative and Direct Methods
   7.6 Multigrid Methods
   Exercises
   References
   Appendix A: Chebyshev Polynomials
   Index