Superconvergence of piecewise linear semi-discretizations for parabolic problems with non-uniform triangulations.

ABSTRACT:
In this paper we study the convergence properties of semi-discrete approximations for parabolic problems on two-dimensional polygonal domains. The semi-discretizations are obtained by using the non-standard piecewise linear finite element method that was introduced by Grigorieff and Ferreira (1998). Main features of that method are the superconvergence on certain non-uniform meshes, as well as that the usual strong coersivity condition of the associated bilinear form is relaxed. Moreover, the method is equivalent to a finite difference scheme that is, in turn, supraconvergent. Here, we will prove that all the properties that are of interest in the stationary case, are also present in the semi-discretizations of parabolic problems.