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Exponential Convergence of hp-FEM for Elliptic Problems in Polyhedra:Mixed Boundary Conditions and Anisotropic Polynomial Degrees

by D. Schötzau and Ch. Schwab

(Report number 2016-05)

Abstract
We prove exponential rates of convergence of \(hp\)-version finite element methods on geometric meshes consisting of hexahedral elements for linear, second-order elliptic boundary-value problems in axiparallel polyhedral domains. We extend and generalize our earlier work for homogeneous Dirichlet boundary conditions and uniform isotropic polynomial degrees to mixed Dirichlet-Neumann boundary conditions and to anisotropic, linearly increasing polynomial degree distributions. In particular, we construct \(H^1\)-conforming quasi-interpolation projectors with exponential consistency bounds on countably normed classes of piecewise analytic functions with singularities at edges, vertices and interfaces of boundary conditions, based on scales of weighted Sobolev norms with non-homogeneous weights in the vicinity of Neumann edges.

Keywords: hp-FEM, second-order elliptic problems in polyhedra, mixed boundary conditions, anisotropic polynomial degrees, exponential convergence

@Techreport{SS16_642,
  author = {D. Sch\"otzau and Ch. Schwab},
  title = {Exponential Convergence of hp-FEM for Elliptic Problems in Polyhedra:Mixed Boundary Conditions and Anisotropic Polynomial Degrees},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-05.pdf },
  year = {2016}
}

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