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.
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.
