The Discontinuous Galerkin (DG) method has been used to solve a wide range of partial differential equations. Especially for advection dominated problems it has proven very reliable and accurate. But even for elliptic problems it has advantages over continuous finite element methods, especially when parallelization and local adaptivity are considered.
In this talk we will first present a variation of the compact DG method for elliptic problems with varying coefficients. For this method we can prove stability on general grids providing a computable bound for all free parameters. We developed this method to solve the compressible Navier-Stokes equations and demonstrated its efficiency in the case of meteorological problems using our implementation within the DUNE software framework, comparing it to the operational code COSMO used by the German weather service.
After introducing the notation and analysis for DG methods in Euclidean spaces, we will present a-priori error estimates for the DG method on surfaces. The surface finite-element method with continuous ansatz functions was analysed a few years ago by Dzuik/Elliot; we extend their results to the interior penalty DG method where the non-smooth approximation of the surface introduces some additional challenges.