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hp-dGFEM for second-order elliptic problems in polyhedra. I: Stability and quasioptimality on geometric meshes

by D. Schötzau and Ch. Schwab and T. Wihler

(Report number 2009-28)

Abstract
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. In order to resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined towards the corresponding neighborhoods. Similarly, the local polynomial degrees are increased s-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, i.e., on $\sigma$-geometric anisotropic meshes of mapped hexahedra with $\kappa$-uniform element mappings and anisotropic polynomial degree distributions of $\mu$-bounded variation. We establish a quasioptimality result that will allow us to prove exponential rates of convergence in the second part of this work.

Keywords:

@Techreport{SSW09_409,
  author = {D. Sch\"otzau and Ch. Schwab and T. Wihler},
  title = {hp-dGFEM for second-order elliptic problems in polyhedra. I: Stability and quasioptimality on geometric meshes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-28.pdf },
  year = {2009}
}

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