On superconvergence results and negative norm estimates for a unidimensional single phase Stefan problem

Abstract

Based on Landau-type transformation, a unidimensional single phase Stefan problem is transformed into a system consisting of parabolic equation with a quadratic nonlinear term and two ordinary differential equations. An H^l-Galerkin method is then applied to estimate the quadratic nonlinear term effectively and optimal estimates in L^∞, L², H¹ and H² norms are obtained without quasiuniformity condition on the finite element mesh. Further using quasiprojection technique, negative norm estimates and superconvergence results are derived. As a result, Galerkin approximation for the free boundary exhibits a superconvergence phenomenon. Since the superconvergence results for the H^l-Galerkin approximations to nonlinear parabolic equations are not available in the literature, the present study has an added significance.