On a Semi Discrete Method for a Class of Parabolic Boundary Value Problems

Abstract

In this paper we discuss a method of discretization in time for a class of evolution problems of the parabolic type. For the sake of generality we work in an abstract, variational setting.

The method, based on a θ-difference scheme, $\theta \in \left[ {\frac{1}{2},1} \right]$, turns out to be a working alternative for the usual Rothe method, explored intensively by e.g. Rektorys and Kačur. Indeed, it provides a constructive method for proving the existence of an exact solution. More important, it may serve quite well as an approximation method. Several results of error estimates have been obtained. In particular the value $\theta = \frac{1}{2}$ is found to give optimal order estimates, if the solution is sufficiently regular.

In most cases a second numerical method will be necessary to solve the elliptic problem at each time point. It is emphasized that a standard Galerkin method can be applied.