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

Authors: Dominik Schötzau, Christoph Schwab and Thomas P. Wihler
Journal: Math. Comp. 85 (2016), 1051-1083
MSC (2010): Primary 65N30
Published electronically: December 11, 2015
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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.

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Additional Information

Dominik Schötzau
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Christoph Schwab
Affiliation: Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland

Thomas P. Wihler
Affiliation: Mathematisches Institut, Universität Bern, 3012 Bern, Switzerland

Keywords: $hp$-dGFEM, second-order elliptic problems in 3D polyhedra, mixed Dirichlet-Neumann boundary conditions, exponential convergence
Received by editor(s): November 8, 2013
Received by editor(s) in revised form: October 16, 2014
Published electronically: December 11, 2015
Additional Notes: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), the European Research Council AdG grant STAHDPDE 247277, and the Swiss National Science Foundation (SNF)
Article copyright: © Copyright 2015 American Mathematical Society