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
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 ?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.
