Exponential convergence for hp-version and spectral finite element methods for elliptic problems in polyhedra

Abstract

We establish exponential convergence of conforming hp-version and spectral finite element methods for second-order, elliptic boundary-value problems with constant coefficients and homogeneous Dirichlet boundary conditions in bounded, axiparallel polyhedra. The source terms are assumed to be piecewise analytic. The conforming hp-approximations are based on σ-geometric meshes of mapped, possibly anisotropic hexahedra and on the uniform and isotropic polynomial degree p ≥ 1. The principal new results are the construction of conforming, patchwise hp-interpolation operators in edge, corner and corner-edge patches which are the three basic building blocks of geometric meshes. In particular, we prove, for each patch type, exponential convergence rates for the H1-norm of the corresponding hp-version (quasi)interpolation errors for functions which belong to a suitable, countably normed space on the patches. The present work extends recent hp-version discontinuous Galerkin approaches to conforming Galerkin finite element methods.