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hp-dGFEM for Second-Order Mixed Elliptic Problems in Polyhedra

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

(Report number 2013-39)

Abstract
We prove exponential rates of convergence of \(hp\)-dG interior penalty (IP) methods for second-order elliptic problems with mixed boundary conditions in polyhedra which are based on axiparallel, \(\sigma\)-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of \(\mu\)-bounded variation. Compared to homogeneous Dirichlet boundary conditions in~\cite{SSWI,SSWII}, for problems with mixed Dirichlet-Neumann boundary conditions, we establish exponential convergence for a nonconforming dG interpolant consisting of elementwise \(L^2\) projections onto elemental polynomial spaces with possibly anisotropic polynomial degrees, and for solutions which belong to a larger analytic class than the solutions considered in \cite{SSWII}. New arguments are introduced for exponential convergence of the dG consistency errors in elements abutting on Neumann edges due to the appearance of non-homogeneous, weighted norms in the analytic regularity at corners and edges. The nonhomogeneous norms entail a reformulation of dG flux terms near Neumann edges, and modification of the stability and quasi-optimality proofs, and the definition of the anisotropic interpolation operators. The exponential convergence results for the piecewise \(L^2\) projection generalizes \cite{SSWI,SSWII} also in the Dirichlet case.

Keywords: $hp$-dGFEM, second-order elliptic problems in 3D polyhedra, mixed Dirichlet-Neumann boundary conditions, exponential convergence

@Techreport{SSW13_536,
  author = {D. Sch\"otzau and Ch. Schwab and T.P. Wihler},
  title = {hp-dGFEM for Second-Order Mixed Elliptic Problems in Polyhedra},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-39.pdf },
  year = {2013}
}

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