hp-dGFEM for second-order mixed elliptic problems in polyhedra

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, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. Compared to homogeneous Dirichlet boundary conditions in [10, 11], for problems with mixed Dirichlet-Neumann boundary conditions, we establish exponential convergence for a nonconforming dG interpolant consisting of elementwise L 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 quasioptimality proofs, and the definition of the anisotropic interpolation operators. The exponential convergence results for the piecewise L projection generalizes [10, 11] also in the Dirichlet case.