Selfadaptive Mesh Modification for Parabolic FBPs: Theory and Computation

Abstract

Finite element approximations of fixed domain formulations of parabolic free boundary problems (FBPs) typically exhibit a suboptimal rate of convergence. For the two-phase Stefan problem, for instance, the order of convergence in L ² for piecewise linear approximations of temperature is never better than linear [13,14], even though one might expect quadratic convergence according to the interpolation theory. Theoretical results are even more pessimistic [5,13,16]. This sort of global numerical pollution is produced by the singularity located on the interface, which is a relevant time dependent unknown on its own right. The main goal is thus to eliminate such a pollution effect by using properly graded meshes. They, in turn, should serve to equidistribute interpolation errors as well as accompany the interface motion. The task of designing an adaptive algorithm, that automatically modifies (or regenerates) a mesh to meet certain predetermined quality criteria, is a challenging one. The aim of this paper is to shed light on several crucial theoretical and computational issues that dictate the quality and efficiency of an adaptive method for FBPs.

This work was partially supported by NSF Grant DMS-8805218 and by MPI (Fondi per la Ricerca Scientifica 40%) and CNR (IAN, Contract 880032601 and Progetto Finalizzato “Sistemi Informatici e Calcolo Parallelo”, Sottoprogetto “Calcolo Scientifico per Grandi Sistemi”) of Italy.