Multilevel Preconditioning for Hybrid Discontinuous Galerkin Methods
Univ.-Prof. Dr. Herbert EggerDec. 21, 2009, 9:30 a.m. T 041
Discontinuous Galerkin methods have become standard tools for discretization of elliptic and hyperbolic pdes. Hybridization allows to overcome some of the deﬃciencies of discontinuous Galerkin methods, such as the increased size and non-optimal sparsity of the global systems.
We investigate multilevel and multigrid solvers for the systems arising from these hybrid discontinuous Galerkin methods. Since the degrees of freedom of a hybrid method are associated to the skeleton of the mesh, the resulting solvers have to be analyzed in the framework of multilevel methods with non-nested spaces. The formulation of appropriate integrid transfer operators, based on local lifting operators, allows to obtain optimality results for the resulting multilevel preconditioners.
We present in detail the analysis for a two level additive Schwarz method, and discuss extensions to multilevel and multigrid methods. The theoretical results are illustrated with numerical examples.