XFEM for the diffusion and sub-diffusion problems in a non-convex domain

Abstract

This paper proposes a space-time finite element method for solving diffusion and sub-diffusion problems in a non-convex domain. This method employs the discontinuous Galerkin (DG) method for temporal discretization and the eXtended Finite Element Method (XFEM) for spatial discretization, which offers higher accuracy than the standard linear finite element method. Sharp error estimates are derived for diffusion and sub-diffusion problems with smooth and non-smooth initial data in a unified approach. Finally, two numerical examples are provided to verify the theoretical results.