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xiv Contents

2.2 The Finite Element Method with Linear Elements . . . . 55
2.3 Stability and Convergence of the
Finite Element Method ... .. .. ... .. .. ... .. 68
2.4 The Implementation of the Finite Element Method:
Part 1 .. ... .. .. ... .. .. ... .. .. ... .. 74
2.5 Solving Sparse Systems of Linear Equations
by Direct Methods .. ... .. .. ... .. .. ... .. 82

3 The Finite Element Method for Linear Elliptic
Boundary Value Problems of Second Order 92
3.1 Variational Equations and Sobolev Spaces . . . . . . . . 92
3.2 Elliptic Boundary Value Problems of Second Order . . . 100
3.3 Element Types and Affine
Equivalent Triangulations . .. .. ... .. .. ... .. 114
3.4 Convergence Rate Estimates .. .. ... .. .. ... .. 131
3.5 The Implementation of the Finite Element Method:
Part 2 .. ... .. .. ... .. .. ... .. .. ... .. 148
3.6 Convergence Rate Results in Case of
Quadrature and Interpolation . .. ... .. .. ... .. 155
3.7 The Condition Number of Finite Element Matrices . . . 163
3.8 General Domains and Isoparametric Elements . . . . . . 167
3.9 The Maximum Principle for Finite Element Methods . . 171

4 Grid Generation and A Posteriori Error Estimation 176
4.1 Grid Generation .. .. ... .. .. ... .. .. ... .. 176
4.2 A Posteriori Error Estimates and Grid Adaptation . . . 185

5 Iterative Methods for Systems of Linear Equations 198
5.1 Linear Stationary Iterative Methods . . . . . . . . . . . . 200
5.2 Gradient and Conjugate Gradient Methods . . . . . . . . 217
5.3 Preconditioned Conjugate Gradient Method . . . . . . . 227
5.4 Krylov Subspace Methods
for Nonsymmetric Systems of Equations .. .. ... .. 233
5.5 The Multigrid Method ... .. .. ... .. .. ... .. 238
5.6 Nested Iterations . .. ... .. .. ... .. .. ... .. 251

6 The Finite Volume Method 255
6.1 The Basic Idea of the Finite Volume Method . . . . . . . 256
6.2 The Finite Volume Method for Linear Elliptic Differen-
tial Equations of Second Order on Triangular Grids . . . 262

7 Discretization Methods for Parabolic Initial Boundary
Value Problems 283
7.1 Problem Setting and Solution Concept . . . . . . . . . . 283
7.2 Semidiscretization by the Vertical Method of Lines . . . 293