hp-DGFEM for second-order mixed elliptic problems polyhedra

Schötzau, Dominik; Schwab, Christoph; Wihler, Thomas (2015). hp-DGFEM for second-order mixed elliptic problems polyhedra. Mathematics of computation, 85(299), pp. 1051-1083. American Mathematical Society 10.1090/mcom/3062

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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, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming 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 on 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 [11, 12] for the pure Dirichlet case.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Wihler, Thomas

Subjects:

500 Science > 510 Mathematics

ISSN:

0025-5718

Publisher:

American Mathematical Society

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

05 Apr 2016 13:47

Last Modified:

31 Mar 2017 08:51

Publisher DOI:

10.1090/mcom/3062

BORIS DOI:

10.7892/boris.77214

URI:

https://boris.unibe.ch/id/eprint/77214

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