hp-dGFEM for Second-Order Mixed Elliptic Problems in Polyhedra: II: Exponential Convergence [report]

Dominik Schötzau, Christoph Schwab, Thomas Pascal Wihler
We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary value problems in three dimensional polyhedral domains. In order to resolve possible corner-, edge-and corneredge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined towards the corresponding neighborhoods. Similarly, the local polynomial degrees are increased s-linearly and possibly
more » ... ally away from singularities. We design interior penalty hp-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed hp-refinements, i.e., on σ-geometric anisotropic meshes of mapped hexahedra with anisotropic polynomial degree distributions of µ-bounded variation. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.
doi:10.3929/ethz-a-010400078 fatcat:zjfnqhr53jbjjeqkssg3m3556u