Numerical Treatment of Partial Differntial Equations

Numerical Treatment of Partial Differential Equations                                                                                                                    Series: Universitext        
Grossmann, Christian, Roos, Hans-Görg, Stynes, Martin
                                                        
Translation and Revision of the 3rd edition of "Numerische Behandlung partieller Differentialgleichungen" published by Teubner, 2005
007, XII, 596 p. 86 illus., Softcover
ISBN: 978-3-540-71582-5
                                                      
                                                                                                                                                                          
This book deals with discretization techniques for partial differential equations of elliptic, parabolic and hyperbolic type. It provides an introduction to the main principles of discretization and gives a presentation of the ideas and analysis of advanced numerical methods in the area. The book is mainly dedicated to finite element methods, but it also discusses difference methods and finite volume techniques.


Coverage offers analytical tools, properties of discretization techniques and hints to algorithmic aspects. It also guides readers to current developments in research with detailed introductions to such recent topics as a posteriori error estimation, discontinuous Galerkin methods or optimal control with partial differential equations. In addition, the authors give basic facts on how to solve generated discrete problems. Chapters on singularly perturbed problems, variational inequalities and optimal control reflect the research interests of the authors.


From the reviews:
"This textbook is the translation and revision of the third German edition of 2005. ... the book deals with different aspects of the numerical solution of elliptic, parabolic and hyperbolic partial differential equations. ... An index and two pages with a summary of the notations used complete the presentation. ... It can be highly recommended for students and engineers but also for numerical analysts." (Riidiger Weiner, Zentralblatt fiir Angewandte Mathematik and Mechanik, Vol. 88 (12), 2008)

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Classification and Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fourier’ s Method, Integral Transforms . . . . . . . . . . . . . . . . . . . . . 5
1.3 Maximum Principle, Fundamental Solution . . . . . . . . . . . . . . . . . 9
1.3.1 Elliptic Boundary Value Problems . . . . . . . . . . . . . . . . . . . 9
1.3.2 Parabolic Equations and Initial-Boundary Value
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Hyperbolic Initial and Initial-Boundary Value Problems 18
2 Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Transportation Problems and Conservation Laws . . . . . . . . . . . . 36
2.3.1 The One-Dimensional Linear Case . . . . . . . . . . . . . . . . . . . 37
2.3.2 Properties of Nonlinear Conservation Laws . . . . . . . . . . . 48
2.3.3 Difference Methods for Nonlinear Conservation Laws . . . 53
2.4 Elliptic Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4.1 Elliptic Boundary Value Problems . . . . . . . . . . . . . . . . . . . 61
2.4.2 The Classical Approach to Finite Difference Methods . . 62
2.4.3 Discrete Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4.4 Difference Stencils and Discretization in General
Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.4.5 Mixed Derivatives, Fourth Order Operators . . . . . . . . . . . 82
2.4.6 Local Grid Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.5 Finite Volume Methods as Finite Difference Schemes . . . . . . . . . 90
2.6 Parabolic Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . 103
2.6.1 Problems in One Space Dimension . . . . . . . . . . . . . . . . . . . 104
2.6.2 Problems in Higher Space Dimensions . . . . . . . . . . . . . . . . 109
2.6.3 Semi-Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.2 Adapted Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.3 Variational Equations and Conforming Approximation . . . . . . . 142
3.4 Weakening V-ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.5 Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.1 A First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.2 Finite-Element-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.2.1 Local and Global Pro5
Element Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
4.5.5 Remarks on Curved Boundaries . . . . . . . . . . . . . . . . . . . . . 254
4.6 Mixed Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
4.6.1 Mixed Variational Equations and Saddle Points . . . . . . . 258
4.6.2 Conforming Approximation of Mixed Variational
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.6.3 Weaker Regularity for the Poisson and Biharmonic
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4.8.3 Error Estimates for a Convection-Diffusion Problem . . . 3024.9 Further Aspects of the Finite Element Method . . . . . . . . . . . . . . 306
4.9.1 Conditioning of the Stiffness Matrix . . . . . . . . . . . . . . . . . 306
4.9.2 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
4.9.3 Superconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
4.9.4 p- and hp-Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
5 Finite Element Methods for Unsteady Problems . . . . . . . . . . . 317
5.1 Parabolic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1.1 On the Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1.2 Semi-Discretization by Finite Elements . . . . . . . . . . . . . . . 321
5.1.3 Temporal Discretization by Standard Methods . . . . . . . . 330
5.1.4 Temporal Discretization with Discontinuous Galerkin
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
5.1.5 Rothe’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
5.1.6 Error Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
5.2 Second-Order Hyperbolic Problems . . . . . . . . . . . . . . . . . . . . . . . . 356
5.2.1 Weak Formulation of the Problem . . . . . . . . . . . . . . . . . . . 356
5.2.2 Semi-Discretization by Finite Elem
6.1.3 Layer-adapted Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
6.2 Parabolic Problems, One-dimensional in Space . . . . . . . . . . . . . . 399
6.2.1 The Analytical Behaviour of the Solution . . . . . . . . . . . . . 399
6.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
6.3 Convection-Diffusion Problems in Several Dimensions . . . . . . . . 406
6.3.1 Analysis of Elliptic Convection-Diffusion Problems . . . . . 406
6.3.2 Discretization on Standard Meshes . . . . . . . . . . . . . . . . . . 412
6.3.3 Layer-adapted Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
6.3.4 Parabolic Problems, Higher-Dimensional in Space . . . . . 430
7 Variational Inequalities, Optimal Control . . . . . . . . . . . . . . . . . . 435
7.1 Analytic Properties
8.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
8.2.1 Gaussian Elimination for Banded Matrices. . . . . . . . . . . . 502
8.2.2 Fast Solution of Discrete Poisson Equations, FFT . . . . . 504

8.3.3 Bl . . . . . . . . . . . . . 538
8.5 Multigrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
8.6 Domain Decomposition, Parallel Algorithms . . . . . . . . . . . . . . . . 560