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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.
List of Symbols
1. Quadratic Functionals on Finite-Dimensional Vector Spaces
1.1 Quadratic Functionals
1.2 The Method of Steepest Descent
1.3 The Conjugate Gradient Method
1.4 The Preconditioned Conjugate Gradient Method
2. Variational Formulation of Boundary Value Problems: Part I
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
3. Variational Formulation of Boundary Value Problems: Part II
3.1 The Concept of Completion
3.2 The Lax-Milgram Lemma and Applications
3.3 Regularity, Symbolic Functions, and Green's Functions
4. The Ritz-Galerkin Method
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
5. The Finite Element Method
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
6. Direct Methods for Solving Finite Element Equations
6.1 Band Matrices
6.2 Direct Methods
6.3 Special Techniques
6.4 Error Analysis
7. Iterative Solution of Finite Element Equations
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
Appendix A: Chebyshev Polynomials
- No. of pages:
- © Academic Press 1984
- 1st August 1984
- Academic Press
- eBook ISBN:
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