hp-dGFEM for Second-Order Mixed Elliptic Problems in Polyhedra

Abstract

We prove exponential rates of convergence of $hp$-dG interior penalty (IP) methods for second-order elliptic problems with mixed boundary conditions in polyhedra which are based on axiparallel, $\sigma$ -geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of $ \mu$-bounded variation. Compared to homogeneous Dirichlet boundary conditions in [10,11], or problems with mixed Dirichlet-Neumann boundary conditions, we establish exponential convergence for a nonconforming dG interpolant consisting of elementwise $L^2$ projections onto elemental polynomial spaces with possibly anisotropic polynomial degrees, and for solutions which belong to a larger analytic class than the solutions considered in [11]. New arguments are introduced for exponential convergence of the dG consistency errors in elements abutting on Neumann edges due to the appearance of non-homogeneous, weighted norms in the analytic regularity at corners and edges. The nonhomogeneous norms entail a reformulation of dG flux terms near Neumann edges, and modification of the stability and quasi-optimality proofs, and the definition of the anisotropic interpolation operators. The exponential convergence results for the piecewise $L^2$ projection generalizes [10,11] also in the Dirichlet case.