hp-dGFEM for Second-Order Elliptic Problems in Polyhedra I: Stability and Quasioptimality on Geometric Meshes

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-, edgeand 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 anisotropically 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 κuniform element mappings and anisotropic polynomial degree distributions of μ-bounded variation. We establish a quasioptimality result that will allow us to prove exponential rates of convergence in the second part of this work.