Found Comput Math (2018) 18:595–660
Exponential Convergence of hp-FEM for Elliptic
Problems in Polyhedra: Mixed Boundary Conditions
and Anisotropic Polynomial Degrees
· Christoph Schwab
Received: 24 January 2016 / Revised: 5 November 2016 / Accepted: 28 January 2017 /
Published online: 13 March 2017
© SFoCM 2017
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 condi-
tions and to anisotropic, which increase linearly over mesh layers away from edges
and vertices. In particular, we construct H
-conforming quasi-interpolation operators
with N degrees of freedom and prove exponential consistency bounds exp(−b
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.
Dedicated to Monique Dauge on the occasion of her 60th birthday.
Communicated by Endre Süli.
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada
(NSERC) and the Swiss National Science Foundation (SNF).
Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver,
BC V6T 1Z2, Canada
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland