Report 1999-04

Time Discretization of Parabolic Problems by the hp-Version of the Discontinuous Galerkin Finite Element Method

D. Schötzau and C. Schwab

Abstract: The Discontinuous Galerkin Finite Element Method (DGFEM) for the time discretization of parabolic problems is analyzed in a hp-version context. Error bounds which are explicit in the time step as well as the approximation order are derived and it is shown that the hp-DGFEM gives spectral convergence in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the hp-DGFEM results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms. For the h-version DGFEM algebraically graded time partitions are determined that give the optimal algebraic convergence rates. A fully discrete hp scheme is discussed exemplarily for the heat equation. The use of certain mesh-design principles for spatial discretizations yields exponential rates of convergence in time and space. Numerical examples confirm the theoretical results.

Keywords: Abstract Parabolic Problems, Discontinuous Galerkin Methods, hp-Version of the Finite Element Method

Paper: Available as PDF (687 KB) or as hardcopy to order reports@sam.math.ethz.ch.

Publishing information: SIAM J. Num. Anal., (2000)