Finite Element Solution of Boundary Value Problems

PrefaceAcknowledgmentsList of Symbols1. 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 References2. 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 References3. 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 References4. 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 References5. 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 References6. 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 References7. 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 ReferencesAppendix A: Chebyshev PolynomialsIndex