Galerkin Finite Element Methods for Parabolic Problems

Vidar Thomée (auth.)9783540331216, 3-540-33121-2

This book surveys the mathematics of Galerkin finite element method as applied to parabolic equations. The approach is to first discretize in the spatial variables by Galerkin’s method, using piecewise polynomial trial functions, and then to apply some time stepping method, most often of single step but also of multistep type, to determine a fully discrete solution. The concern is stability and error analysis of approximate solutions in various norms, and under various regularity assumptions on the exact solution. The book aims at giving insight in current ideas and methods of analysis rather than at pursuing each approach to its limit. It is essentially self-contained, and the use of simple model situations makes it easily accessible also for beginners in the field.

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Table of contents :
Front Matter….Pages I-X
The Standard Galerkin Method….Pages 1-22
Methods Based on More General Approximations of the Elliptic Problem….Pages 23-34
Nonsmooth Data Error Estimates….Pages 35-50
More General Parabolic Equations….Pages 51-62
Maximum-Norm Stability and Error Estimates….Pages 63-80
Negative Norm Estimates and Superconvergence….Pages 81-94
Single Step Fully Discrete Schemes for the Homogeneous Equation….Pages 95-110
Single Step Methods and Rational Approximations of Semigroups….Pages 111-125
Single Step Fully Discrete Schemes for the Inhomogeneous Equation….Pages 127-144
Multistep Backward Difference Methods….Pages 145-162
Incomplete Iterative Solution of the Algebraic Systems at the Time Levels….Pages 163-179
The Discontinuous Galerkin Time Stepping Method….Pages 181-208
A Nonlinear Problem….Pages 209-222
Semilinear Parabolic Equations….Pages 223-238
The Method of Lumped Masses….Pages 239-252
The H 1 and H -1 Methods….Pages 253-266
A Mixed Method….Pages 267-278
A Singular Problem….Pages 279-287
Back Matter….Pages 289-302