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

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 con rm the theoretical results.

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

“…For Examples 2 and 3, due to the singular behavior of the solution at the start, the time partition is geometrically refined at t = 0 (note that we have used a logarithmic scale on the horizontal axis). This is in correspondence with the a priori error analysis in [24]. Furthermore, we notice that the polynomial degrees tend to decrease away from t = 0, which is due to the fact that the right-hand side of (40) is zero, and hence, the solution is flattened out quickly.…”

“…In this section, we shall recall and discuss the hp-version formulations of the continuous and discontinuous Galerkin time-stepping methods from [23,24] and [31], respectively, for the time (semi-) discretization of the parabolic problem (1).…”

“…The dG method in (5) is consistent and has a unique solution; see [28,Chapter 12] or [23,24]. Similarly to the cG method, it can also be interpreted as an implicit one-step time-stepping scheme.…”

“…For detailed implementational techniques for hp-version dG time-stepping methods we refer to [24,29]. We discretize our model problems using a high-order finite element method in space so that the spatial errors are dominated by the errors resulting from the cG and dG time discretizations.…”

“…The main feature of these hp-methods is their ability to approximate smooth solutions-with possible local singularities-at high algebraic or even exponential rates of converge. In particular, exponential convergence can be achieved in the numerical approximation of problems with start-up singularities; see [24,25,30] for linear parabolic PDEs and [4] for Volterra integro-differential equations with weakly singular kernels. Furthermore, it has been proved in [20,21] that the combination of hp-version time-stepping with suitable wavelet spatial discretizations results in log-linear complexity solution algorithms for nonlocal evolution processes involving pseudo-differential operators.…”