Galerkin Finite Element Methods for Parabolic Problems

1. Front Matter

   Pages I-X

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2. The Standard Galerkin Method

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   Pages 1-22

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

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   Pages 23-34

4. Nonsmooth Data Error Estimates

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   Pages 35-50

5. More General Parabolic Equations

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   Pages 51-62

6. Maximum-Norm Stability and Error Estimates

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   Pages 63-80

7. Negative Norm Estimates and Superconvergence

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   Pages 81-94

8. Single Step Fully Discrete Schemes for the Homogeneous Equation

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   Pages 95-110

9. Single Step Methods and Rational Approximations of Semigroups

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   Pages 111-125

10. Single Step Fully Discrete Schemes for the Inhomogeneous Equation

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    Pages 127-144

11. Multistep Backward Difference Methods

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    Pages 145-162

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

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    Pages 163-179

13. The Discontinuous Galerkin Time Stepping Method

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    Pages 181-208

14. A Nonlinear Problem

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    Pages 209-222

15. Semilinear Parabolic Equations

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    Pages 223-238

16. The Method of Lumped Masses

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    Pages 239-252

17. The H ¹ and H ^-1 Methods

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    Pages 253-266

18. A Mixed Method

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    Pages 267-278

19. A Singular Problem

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    Pages 279-287

20. Back Matter

    Pages 289-302

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My purpose in this monograph is to present an essentially self-contained account of the mathematical theory of Galerkin finite element methods as applied to parabolic partial differential equations. The emphases and selection of topics reflects my own involvement in the field 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 finite 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 doing so 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 first treating the time discretization of an abstract differential equation in a Hilbert space setting, and the chapter on the discontinuous Galerkin method has been completely rewritten.

• Approximation
• Differential Equations
• Finite Element Theory
• Galerkin Methods
• Parabolic Partial
• algebra
• differential equation
• finite element method
• mathematics
• maximum

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

  Vidar Thomée

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