Author: Andreas Dedner, Assistant Professor, Department of applied mathematics, University of Freiburg

Title: hp Adaptive Stabilization of the Discontinuous Galerkin Method for Evolution Equations

Abstract:

In this talk we present an hp-adaptive scheme in space and time for the discretization of systems of general convection-diffusion equations with source term. We base our method on the higher order Discontinuous Galerkin method in space and explicit methods in time using general parallel grid structures with h-adaptivity. Our focus is on the convection dominated case, so we discuss approaches for gradient limiting and p-adaptivity for stabilizing the scheme in the regions of strong gradients or discontinuities. The basis of the hp-adptivity is an a-posteriori error estimate for the semi-discrete method. By combining several "simple" spatial operators it is possible to use our method to solve quite complex problems.

For the implementation of the scheme we use the free software package DUNE (dune.mathematik.uni-freiburg.de). In this package, general interfaces for grid-based numerical methods are described and a variety of grid structures and numerical schemes are implemented. The interface-based approach allows the coding of numerical schemes independent of the grid structure and the spatial dimension. Parallelization, load balancing, and grid adaptation are part of the general interface and can thus be easily used in the numerical scheme. Our approach uses modern software design in C++ to combine flexibility with high efficiency