Thomee Galerkin Finite Element Methods for Parabolic Problems

2. Auflage 2007
ISBN: 978-3-540-33122-3
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 364 Seiten

My purpose in this monograph is to present an essentially self-contained account of the mathematical theory of Galerkin ?nite element methods as appliedtoparabolicpartialdi?erentialequations. Theemphasesandselection of topics re?ects my own involvement in the ?eld over the past 25 years, and my ambition has been to stress ideas and methods of analysis rather than to describe the most general and farreaching results possible. Since the formulation and analysis of Galerkin ?nite element methods for parabolic problems are generally based on ideas and results from the corresponding theory for stationary elliptic problems, such material is often included in the presentation. The basis of this work is my earlier text entitled Galerkin Finite Element Methods for Parabolic Problems, Springer Lecture Notes in Mathematics, No. 1054, from 1984. This has been out of print for several years, and I have felt a need and been encouraged by colleagues and friends to publish an updated version. In doingso I have included most of the contents of the 14 chapters of the earlier work in an updated and revised form, and added four new chapters, on semigroup methods, on multistep schemes, on incomplete iterative solution of the linear algebraic systems at the time levels, and on semilinear equations. The old chapters on fully discrete methods have been reworked by ?rst treating the time discretization of an abstract di?erential equation in a Hilbert space setting, and the chapter on the discontinuous Galerkin method has been completely rewritten.

Autoren/Hrsg.

Thomee, Vidar

Inhaltsverzeichnis

1;Preface;5
2;Preface to the Second Edition;9
3;Table of Contents;11
4;1. The Standard Galerkin Method;13
5;2. Methods Based on More General Approximations of the Elliptic Problem;37
6;3. Nonsmooth Data Error Estimates;48
7;4. More General Parabolic Equations;66
8;5. Negative Norm Estimates and Superconvergence;78
9;6. Maximum-Norm Estimates and Analytic Semigroups;92
10;7. Single Step Fully Discrete Schemes for the Homogeneous Equation;121
11;8. Single Step Fully Discrete Schemes for the Inhomogeneous Equation;138
12;9. Single Step Methods and Rational Approximations of Semigroups;157
13;10. Multistep Backward Difference Methods;170
14;11. Incomplete Iterative Solution of the Algebraic Systems at the Time Levels;192
15;12. The Discontinuous Galerkin Time Stepping Method;210
16;13. A Nonlinear Problem;238
17;14. Semilinear Parabolic Equations;252
18;15. The Method of Lumped Masses;268
19;16. The H1 and H- Methods;286
20;17. A Mixed Method;300
21;18. A Singular Problem;312
22;19. Problems in Polygonal Domains;323
23;20. Time Discretization by Laplace Transformation and Quadrature;341
24;References;361
25;Index;375