Exponential Convergence of hp -DGFEM for Elliptic Problems in Polyhedral Domains

Abstract

We review the recent results of D. Schotzau et al. (hp-dGFEM for elliptic problems in polyhedra. I: Stability and quasioptimality on geometric meshes. Technical report 2009-28, Seminar for applied mathematics, ETH Zurich, 2009. To appear in SIAM J Numer Anal, 2013; hp-dGFEM for elliptic problems in polyhedra. II: Exponential convergence. Technical report 2009-29, Seminar for applied mathematics, ETH Zurich, 2009. To appear in SIAM J Numer Anal, 2013), and establish the exponential convergence of hp-version discontinuous Galerkin finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and constant coefficients in three-dimsional and axiparallel polyhedra. The exponential rates are confirmed in a series of numerical tests.