hp Discontinuous Galerkin Time Stepping for Parabolic Problems

Abstract

The algorithmic pattern of the hp Discontinuous Galerkin Finite Element Method (DGFEM) for the time semidiscretization of abstract parabolic evolution equations is presented. In combination with a continuous $hp$ discretization in space we present a fully discrete hp-scheme for the numerical solution of parabolic problems. Numerical examples for the heat equation in a two dimensional domain confirm the exponential convergence rates which are predicted by theoretical results, under realistic assumptions on the initial condition and the data. Different methods to reduce the computational cost of the DGFEM are compared.