Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions

Abstract

This work presents novel finite element approaches for solving a parabolic partial differential equation with discontinuous coefficients and low regularity solutions in a bounded convex polyhedral domain. A spatial semi-discretization based on symmetric interior penalty Galerkin (SIPG) approximations is constructed and analyzed by using discontinuous piecewise linear functions. For smooth initial data, spatial errors in the broken L2, H1 and L2(H1) norms are proven to be optimal with respect to low regularity solutions, which are only piecewise H1+s smooth with 0<s≤1. Furthermore we present the full SIPG discretizations based on an Euler backward finite difference time discretization. The proposed approximations are shown to be unconditionally stable, and have nearly the optimal L2(L2) and L2(H1) error estimates, even when the regularities of the solutions are low on the whole domain. The error estimates are optimal with respect to time, sharply depending on the indexes in the global and local regularity. Numerical experiments for two-dimensional parabolic interface problems verify the theoretical convergence rates.