Exponential Convergence of hp-FEM for Elliptic Problems in Polyhedra: Mixed Boundary Conditions and Anisotropic Polynomial Degrees

Abstract

We prove exponential rates of convergence of hp-version finite element methods on geometric meshes consisting of hexahedral elements for linear, second-order elliptic boundary value problems in axiparallel polyhedral domains. We extend and generalize our earlier work for homogeneous Dirichlet boundary conditions and uniform isotropic polynomial degrees to mixed Dirichlet–Neumann boundary conditions and to anisotropic, which increase linearly over mesh layers away from edges and vertices. In particular, we construct $$H^1$$ -conforming quasi-interpolation operators with N degrees of freedom and prove exponential consistency bounds $$\exp (-b\root 5 \of {N})$$ for piecewise analytic functions with singularities at edges, vertices and interfaces of boundary conditions, based on countably normed classes of weighted Sobolev spaces with non-homogeneous weights in the vicinity of Neumann edges.