Time discretization of parabolic problems by the HP-version of the discontinuous Galerkin finite element method

The discontinuous Galerkin finite element method (DGFEM) for the time discr etization of parabolic problems is analyzed in the context of the hp-versio n of the Galerkin method. Error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that th e hp-DGFEM gives spectral convergence in problems with smooth time dependen ce. In conjunction with geometric time partitions it is proved that the hp- DGFEM results in exponential rates of convergence for piecewise analytic so lutions exhibiting singularities induced by incompatible initial data or pi ecewise analytic forcing terms. For the h-version DGFEM algebraically grade d time partitions are determined that give the optimal algebraic convergenc e rates. fully discrete hp scheme is discussed exemplarily for the heat equ ation. The use of certain mesh-design principles for the spatial discretiza tions yields exponential rates of convergence in time and space. Numerical examples con rm the theoretical results.