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$hp$-dGFEM for Second-Order Elliptic Problems in Polyhedra I: Stability on Geometric Meshes

2013; Society for Industrial and Applied Mathematics; Volume: 51; Issue: 3 Linguagem: Inglês

10.1137/090772034

1095-7170

Dominik Schötzau, Ch. Schwab, Thomas P. Wihler,

Advanced Mathematical Modeling in Engineering

We introduce and analyze $hp$-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. We design interior penalty $hp$-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed $hp$-refinements. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.