Numerical Methods in Engineering With MATLAB

Numerical Methods for Elliptic and Parabolic Partial Differential Equations......Page 1
Series Preface......Page 6
Preface to the English Edition......Page 8
Preface to the German Edition......Page 12
Contents......Page 14
0.1 The Basic Partial Di.erential Equation Models......Page 18
0.2 Reactions and Transport in Porous Media......Page 22
0.3 Fluid Flow in PorousMedia......Page 24
0.4 Reactive Solute Transport in Porous Media......Page 28
0.5 Boundary and Initial Value Problems......Page 31
1.1 The Dirichlet Problem for the Poisson Equation......Page 36
1.2 The Finite Di.erenceMethod......Page 38
1.3 Generalizations and Limitations of the Finite Di.erenceMethod......Page 46
1.4 Maximum Principles and Stability......Page 53
2.1 Variational Formulation for the Model Problem......Page 63
2.2 The Finite Element Method with Linear Elements......Page 72
2.3 Stability and Convergence of the Finite ElementMethod......Page 85
2.4 The Implementation of the Finite Element Method: Part 1......Page 91
2.5 Solving Sparse Systems of Linear Equations by DirectMethods......Page 99
3.1 Variational Equations and Sobolev Spaces......Page 109
3.2 Elliptic Boundary Value Problems of Second Order......Page 117
3.3 Element Types and A.ne Equivalent Triangulations......Page 131
3.4 ConvergenceRate Estimates......Page 148
3.5 The Implementation of the Finite Element Method: Part 2......Page 165
3.6 Convergence Rate Results in Case of Quadrature and Interpolation......Page 172
3.7 The Condition Number of Finite Element Matrices......Page 180
3.8 General Domains and Isoparametric Elements......Page 184
3.9 The Maximum Principle for Finite Element Methods......Page 188
4.1 Grid Generation......Page 193
4.2 A Posteriori Error Estimates and Grid Adaptation......Page 202
5 Iterative Methods for Systems of Linear Equations......Page 215
5.1 Linear Stationary Iterative Methods......Page 217
5.2 Gradient and Conjugate Gradient Methods......Page 234
5.3 Preconditioned Conjugate Gradient Method......Page 244
5.4 Krylov Subspace Methods for Nonsymmetric Systems of Equations......Page 250
5.5 TheMultigridMethod......Page 255
5.6 Nested Iterations......Page 268
6 The Finite Volume Method......Page 272
6.1 The Basic Idea of the Finite Volume Method......Page 273
6.2 The Finite Volume Method for Linear Elliptic Di.erential Equations of Second Order on Triangular Grids......Page 279
7.1 Problem Setting and Solution Concept......Page 300
7.2 Semidiscretization by the Vertical Method of Lines......Page 310
7.3 Fully Discrete Schemes......Page 328
7.4 Stability......Page 332
7.5 The Maximum Principle for the One-Step-ThetaMethod......Page 340
7.6 Order of Convergence Estimates......Page 347
8 Iterative Methods for Nonlinear Equations......Page 359
8.1 Fixed-Point Iterations......Page 361
8.2 Newton\'s Method and Its Variants......Page 365
8.3 Semilinear Boundary Value Problems for Elliptic and Parabolic Equations......Page 377
9.1 Standard Methods and Convection-Dominated Problems......Page 385
9.2 The Streamline-Di.usion Method......Page 392
9.3 Finite VolumeMethods......Page 400
9.4 The Lagrange-Galerkin Method......Page 404
A.1 Notation......Page 407
A.2 Basic Concepts of Analysis......Page 410
A.3 Basic Concepts of Linear Algebra......Page 411
A.4 Some De.nitions and Arguments of Linear Functional Analysis......Page 416
A.5 Function Spaces......Page 421
References: Textbooks and Monographs......Page 426
References: Journal Papers......Page 429
Index......Page 432