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Exponential Convergence for hp-Version and Spectral Finite Element Methods for Elliptic Problems in Polyhedra

by D. Schötzau and Ch. Schwab

(Report number 2014-38)

Abstract
We establish exponential convergence of conforming \(hp\)-version and spectral finite element methods for second-order, elliptic boundary-value problems with constant coefficients and homogeneous Dirichlet boundary conditions in bounded, axiparallel polyhedra. The source terms are assumed to be piecewise analytic. The conforming \(hp\)-approximations are based on \(\sigma\)-geometric meshes of mapped, possibly anisotropic hexahedra and on the uniform and isotropic polynomial degree~\(p \geq 1\). The principal new results are the construction of conforming, patchwise \(hp\)-interpolation operators in edge, corner and corner-edge patches which are the three basic building blocks of geometric meshes. In particular, we prove, for each patch type, exponential convergence rates for the \(H^1\)-norm of the corresponding \(hp\)-version (quasi)interpolation errors for functions which belong to a suitable, countably normed space on the patches. The present work extends the discontinuous Galerkin approaches in~\cite{SSWI,SSWII} to conforming \(hp\)-Galerkin finite element methods.

Keywords: hp-FEM, spectral FEM, second-order elliptic problems in polyhedra, exponential convergence

@Techreport{SS14_588,
  author = {D. Sch\"otzau and Ch. Schwab},
  title = {Exponential Convergence for hp-Version and Spectral Finite Element Methods for Elliptic Problems in Polyhedra},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-38},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-38.pdf },
  year = {2014}
}

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