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

Abstract:

We prove exponential rates of convergence of $hp$-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, $\sigma$-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of $\mu$-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in our earlier works (2013). For such solutions, we establish the exponential convergence of a non-conforming dG interpolant given by local $L^2$-projections on elements away from corners and edges and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of our earlier works (2013) for the pure Dirichlet case.