Galerkin Finite Element Methods for Parabolic Problems

✍ Scribed by Vidar Thomée (auth.)

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 364
• Edition: 2
• Category: Library

✦ Synopsis

This book provides insight in the mathematics of Galerkin finite element method as applied to parabolic equations. The approach is based on first discretizing in the spatial variables by Galerkin's method, using piecewise polynomial trial functions, and then applying some single step or multistep time stepping method. The concern is stability and error analysis of approximate solutions in various norms, and under various regularity assumptions on the exact solution. The book gives an excellent insight in the present ideas and methods of analysis. The second edition has been influenced by recent progress in application of semigroup theory to stability and error analysis, particulatly in maximum-norm. Two new chapters have also been added, dealing with problems in polygonal, particularly noncovex, spatial domains, and with time discretization based on using Laplace transformation and quadrature.

✦ Table of Contents

The Standard Galerkin Method....Pages 1-24
Methods Based on More General Approximations of the Elliptic Problem....Pages 25-35
Nonsmooth Data Error Estimates....Pages 37-54
More General Parabolic Equations....Pages 55-66
Negative Norm Estimates and Superconvergence....Pages 67-80
Maximum-Norm Estimates and Analytic Semigroups....Pages 81-109
Single Step Fully Discrete Schemes for the Homogeneous Equation....Pages 111-127
Single Step Fully Discrete Schemes for the Inhomogeneous Equation....Pages 129-147
Single Step Methods and Rational Approximations of Semigroups....Pages 149-161
Multistep Backward Difference Methods....Pages 163-184
Incomplete Iterative Solution of the Algebraic Systems at the Time Levels....Pages 185-202
The Discontinuous Galerkin Time Stepping Method....Pages 203-230
A Nonlinear Problem....Pages 231-244
Semilinear Parabolic Equations....Pages 245-260
The Method of Lumped Masses....Pages 261-278
The H 1 and H −1 Methods....Pages 279-292
A Mixed Method....Pages 293-304
A Singular Problem....Pages 305-315
Problems in Polygonal Domains....Pages 317-334
Time Discretization by Laplace Transformation and Quadrature....Pages 335-354

✦ Subjects

Mathematical and Computational Physics