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Galerkin Finite Element Methods for Parabolic Problems

  • Book
  • © 2006

Overview

Authors:
  1. Vidar Thomée

    1. Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden

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Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 25)

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Table of contents (20 chapters)

  1. Front Matter

    Pages I-XII
    Download chapter PDF

  2. The Standard Galerkin Method

    • Vidar Thomée
    Pages 1-24

  3. Methods Based on More General Approximations of the Elliptic Problem

    • Vidar Thomée
    Pages 25-35

  4. Nonsmooth Data Error Estimates

    • Vidar Thomée
    Pages 37-54

  5. More General Parabolic Equations

    • Vidar Thomée
    Pages 55-66

  6. Negative Norm Estimates and Superconvergence

    • Vidar Thomée
    Pages 67-80

  7. Maximum-Norm Estimates and Analytic Semigroups

    • Vidar Thomée
    Pages 81-109

  8. Single Step Fully Discrete Schemes for the Homogeneous Equation

    • Vidar Thomée
    Pages 111-127

  9. Single Step Fully Discrete Schemes for the Inhomogeneous Equation

    • Vidar Thomée
    Pages 129-147

  10. Single Step Methods and Rational Approximations of Semigroups

    • Vidar Thomée
    Pages 149-161

  11. Multistep Backward Difference Methods

    • Vidar Thomée
    Pages 163-184

  12. Incomplete Iterative Solution of the Algebraic Systems at the Time Levels

    • Vidar Thomée
    Pages 185-202

  13. The Discontinuous Galerkin Time Stepping Method

    • Vidar Thomée
    Pages 203-230

  14. A Nonlinear Problem

    • Vidar Thomée
    Pages 231-244

  15. Semilinear Parabolic Equations

    • Vidar Thomée
    Pages 245-260

  16. The Method of Lumped Masses

    • Vidar Thomée
    Pages 261-278

  17. The H1 and H−1 Methods

    • Vidar Thomée
    Pages 279-292

  18. A Mixed Method

    • Vidar Thomée
    Pages 293-304

  19. A Singular Problem

    • Vidar Thomée
    Pages 305-315

  20. Problems in Polygonal Domains

    • Vidar Thomée
    Pages 317-334
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden

    Vidar Thomée

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