Optimal preconditioners of spectral Discontinuous Galerkin methods for elliptic boundary value problems

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Fully flexible hp-Discontinuous Galerkin (DG) schemes in the sense that arbitrary local mesh refinements and arbitrary polynomial degree distributions are permitted offer a powerful discretization platform for a wide scope of partial differential equations. For elliptic boundary value problems corresponding linear systems quickly become very ill-conditioned. So far, efficient preconditioners seem to be known only under strong restrictions on meshes and degree distributions thereby severely constraining the full DG potential. This project aims at developing efficient preconditioners for the „fully flexible“ case. The auxiliary space method, in a catenated form, is used as the conceptual platform. A major new ingredient are certain hierarchies of low order finite element spaces on anisotropic dyadic meshes associated with non-nested Legendre-Gauß-Lobatto grids that offer in our opinion a chance to overcome the main obstructions to the currently used methodologies. The development of corresponding theoretical foundations is to be complemented by an appropriate software concept.Finally, the connection with recent DG hybridization techniques will be explored.