Deuflhard / Weiser Adaptive Numerical Solution of PDEs

1. Auflage 2012
ISBN: 978-3-11-028311-2
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 436 Seiten

Reihe: De Gruyter Textbook

This book deals with the general topic “Numerical solution of partial differential equations (PDEs)” with a focus on adaptivity of discretizations in space and time. By and large, introductory textbooks like “Numerical Analysis in Modern Scientific Computing” by Deuflhard and Hohmann should suffice as a prerequisite. The emphasis lies on elliptic and parabolic systems. Hyperbolic conservation laws are treated only on an elementary level excluding turbulence. Numerical Analysis is clearly understood as part of Scientific Computing. The focus is on the efficiency of algorithms, i.e. speed, reliability, and robustness, which directly leads to the concept of adaptivity in algorithms. The theoretical derivation and analysis is kept as elementary as possible. Nevertheless required somewhat more sophisticated mathematical theory is summarized in comprehensive form in an appendix. Complex relations are explained by numerous figures and illustrating examples. Non-trivial problems from regenerative energy, nanotechnology, surgery, and physiology are inserted. The text will appeal to graduate students and researchers on the job in mathematics, science, and technology. Conceptually, it has been written as a textbook including exercises and a software list, but at the same time it should be well-suited for self-study.

Zielgruppe

Graduate Students, PhD Students, and Lecturers in Mathematics, Physics, Engineering Sciences, Scientific Computing, and Applied Mathematics; Academic Libraries

Autoren/Hrsg.

Deuflhard, Peter

Weiser, Martin

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen

Inhaltsverzeichnis

1;Preface;5
2;Outline;13
3;1 Elementary Partial Differential Equations;17
3.1;1.1 Laplace and Poisson Equation;17
3.1.1;1.1.1 Boundary Value Problems;18
3.1.2;1.1.2 Initial Value Problem;22
3.1.3;1.1.3 Eigenvalue Problem;24
3.2;1.2 Diffusion Equation;27
3.3;1.3 Wave Equation;30
3.4;1.4 Schrödinger Equation;35
3.5;1.5 Helmholtz Equation;38
3.5.1;1.5.1 Boundary Value Problems;38
3.5.2;1.5.2 Time-harmonic Differential Equations;39
3.6;1.6 Classification;41
3.7;1.7 Exercises;43
4;2 Partial Differential Equations in Science and Technology;46
4.1;2.1 Electrodynamics;46
4.1.1;2.1.1 Maxwell Equations;46
4.1.2;2.1.2 Optical Model Hierarchy;49
4.2;2.2 Fluid Dynamics;52
4.2.1;2.2.1 Euler Equations;53
4.2.2;2.2.2 Navier-Stokes Equations;56
4.2.3;2.2.3 Prandtl’s Boundary Layer;61
4.2.4;2.2.4 Porous Media Equation;63
4.3;2.3 Elastomechanics;64
4.3.1;2.3.1 Basic Concepts of Nonlinear Elastomechanics;64
4.3.2;2.3.2 Linear Elastomechanics;68
4.4;2.4 Exercises;71
5;3 Finite Difference Methods for Poisson Problems;74
5.1;3.1 Discretization of Standard Problem;74
5.1.1;3.1.1 Discrete Boundary Value Problems;75
5.1.2;3.1.2 Discrete Eigenvalue Problem;80
5.2;3.2 Approximation Theory on Uniform Grids;83
5.2.1;3.2.1 Discretization Error in L2;85
5.2.2;3.2.2 Discretization Error in L?;88
5.3;3.3 Discretization on Nonuniform Grids;90
5.3.1;3.3.1 One-dimensional Special Case;90
5.3.2;3.3.2 Curved Boundaries;92
5.4;3.4 Exercises;95
6;4 Galerkin Methods;98
6.1;4.1 General Scheme;98
6.1.1;4.1.1 Weak Solutions;98
6.1.2;4.1.2 Ritz Minimization for Boundary Value Problems;101
6.1.3;4.1.3 Rayleigh-Ritz Minimization for Eigenvalue Problems;105
6.2;4.2 Spectral Methods;107
6.2.1;4.2.1 Realization by Orthogonal Systems;108
6.2.2;4.2.2 Approximation Theory;112
6.2.3;4.2.3 Adaptive Spectral Methods;115
6.3;4.3 Finite Element Methods;120
6.3.1;4.3.1 Meshes and Finite Element Spaces;120
6.3.2;4.3.2 Elementary Finite Elements;123
6.3.3;4.3.3 Realization of Finite Elements;133
6.4;4.4 Approximation Theory for Finite Elements;140
6.4.1;4.4.1 Boundary Value Problems;140
6.4.2;4.4.2 Eigenvalue Problems;143
6.4.3;4.4.3 Angle Condition for Nonuniform Meshes;148
6.5;4.5 Exercises;151
7;5 Numerical Solution of Linear Elliptic Grid Equations;155
7.1;5.1 Direct Elimination Methods;156
7.1.1;5.1.1 Symbolic Factorization;157
7.1.2;5.1.2 Frontal Solvers;159
7.2;5.2 Matrix Decomposition Methods;162
7.2.1;5.2.1 Jacobi Method;164
7.2.2;5.2.2 Gauss-Seidel Method;166
7.3;5.3 Conjugate Gradient Method;168
7.3.1;5.3.1 CG-Method as Galerkin Method;168
7.3.2;5.3.2 Preconditioning;171
7.3.3;5.3.3 Adaptive PCG-method;175
7.3.4;5.3.4 A CG-variant for Eigenvalue Problems;177
7.4;5.4 Smoothing Property of Iterative Solvers;182
7.4.1;5.4.1 Illustration for the Poisson Model Problem;182
7.4.2;5.4.2 Spectral Analysis for Jacobi Method;186
7.4.3;5.4.3 Smoothing Theorems;187
7.5;5.5 Iterative Hierarchical Solvers;192
7.5.1;5.5.1 Classical Multigrid Methods;194
7.5.2;5.5.2 Hierarchical-basis Method;202
7.5.3;5.5.3 Comparison with Direct Hierarchical Solvers;205
7.6;5.6 Power Optimization of a Darrieus Wind Generator;206
7.7;5.7 Exercises;212
8;6 Construction of Adaptive Hierarchical Meshes;215
8.1;6.1 A Posteriori Error Estimators;215
8.1.1;6.1.1 Residual Based Error Estimator;218
8.1.2;6.1.2 Triangle Oriented Error Estimators;223
8.1.3;6.1.3 Gradient Recovery;227
8.1.4;6.1.4 Hierarchical Error Estimators;231
8.1.5;6.1.5 Goal-oriented Error Estimation;234
8.2;6.2 Adaptive Mesh Refinement;235
8.2.1;6.2.1 Equilibration of Local Discretization Errors;236
8.2.2;6.2.2 Refinement Strategies;241
8.2.3;6.2.3 Choice of Solvers on Adaptive Hierarchical Meshes;245
8.3;6.3 Convergence on Adaptive Meshes;245
8.3.1;6.3.1 A Convergence Proof;246
8.3.2;6.3.2 An Example with a Reentrant Corner;248
8.4;6.4 Design of a Plasmon-Polariton Waveguide;252
8.5;6.5 Exercises;256
9;7 Adaptive Multigrid Methods for Linear Elliptic Problems;258
9.1;7.1 Subspace Correction Methods;258
9.1.1;7.1.1 Basic Principle;259
9.1.2;7.1.2 Sequential Subspace Correction Methods;262
9.1.3;7.1.3 Parallel Subspace Correction Methods;267
9.1.4;7.1.4 Overlapping Domain Decomposition Methods;271
9.1.5;7.1.5 Higher-order Finite Elements;278
9.2;7.2 Hierarchical Space Decompositions;283
9.2.1;7.2.1 Decomposition into Hierarchical Bases;284
9.2.2;7.2.2 L2-orthogonal Decomposition: BPX;290
9.3;7.3 Multigrid Methods Revisited;294
9.3.1;7.3.1 Additive Multigrid Methods;294
9.3.2;7.3.2 Multiplicative Multigrid Methods;298
9.4;7.4 Cascadic Multigrid Methods;301
9.4.1;7.4.1 Theoretical Derivation;301
9.4.2;7.4.2 Adaptive Realization;307
9.5;7.5 Eigenvalue Problem Solvers;312
9.5.1;7.5.1 Linear Multigrid Method;313
9.5.2;7.5.2 Rayleigh Quotient Multigrid Method;315
9.6;7.6 Exercises;318
10;8 Adaptive Solution of Nonlinear Elliptic Problems;322
10.1;8.1 Discrete Newton Methods for Nonlinear Grid Equations;323
10.1.1;8.1.1 Exact Newton Methods;324
10.1.2;8.1.2 Inexact Newton-PCG Methods;328
10.2;8.2 Inexact Newton-Multigrid Methods;331
10.2.1;8.2.1 Hierarchical Grid Equations;331
10.2.2;8.2.2 Realization of Adaptive Algorithm;333
10.2.3;8.2.3 An Elliptic Problem Without a Solution;337
10.3;8.3 Operation Planning in Facial Surgery;340
10.4;8.4 Exercises;343
11;9 Adaptive Integration of Parabolic Problems;345
11.1;9.1 Time Discretization of Stiff Differential Equations;345
11.1.1;9.1.1 Linear Stability Theory;346
11.1.2;9.1.2 Linearly Implicit One-step Methods;352
11.1.3;9.1.3 Order Reduction;359
11.2;9.2 Space-time Discretization of Parabolic PDEs;365
11.2.1;9.2.1 Adaptive Method of Lines;366
11.2.2;9.2.2 Adaptive Method of Time Layers;374
11.2.3;9.2.3 Goal-oriented Error Estimation;383
11.3;9.3 Electrical Excitation of the Heart Muscle;386
11.3.1;9.3.1 Mathematical Models;386
11.3.2;9.3.2 Numerical Simulation;387
11.4;9.4 Exercises;390
12;A Appendix;392
12.1;A.1 Fourier Analysis and Fourier Transform;392
12.2;A.2 Differential Operators in ?3;393
12.3;A.3 Integral Theorems;395
12.4;A.4 Delta-Distribution and Green's Functions;399
12.5;A.5 Sobolev Spaces;404
12.6;A.6 Optimality Conditions;409
13;B Software;410
13.1;B.1 Adaptive Finite Element Codes;410
13.2;B.2 Direct Solvers;411
13.3;B.3 Nonlinear Solvers;411
14;Bibliography;413
15;Index;427

Über Autor(innen)

Peter Deuflhard and Martin Weiser, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany.