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Symmetric Interior penalty discontinuous Galerkin methods for elliptic problems in polygons

by F. Müller and D. Schötzau and Ch. Schwab

(Report number 2017-15)

Abstract
We analyze symmetric interior penalty discontinuous Galerkin finite element methods for linear, second-order elliptic boundary-value problems in polygonal domains \(\Omega\) where solutions exhibit singular behavior near corners. To resolve corner singularities, we admit both, graded meshes and bisection refinement meshes. We prove that judiciously chosen refinement parameters in these mesh families imply optimal asymptotic rates of convergence with respect to the total number of degrees of freedom~\(N\), both for the DG energy norm error and the \(L^2\)-norm error. The sharpness of our asymptotic convergence rate estimates is confirmed in a series of numerical experiments.

Keywords: Elliptic Boundary-Value Problems, Finite Element Methods, Discontinuous Galerkin Methods, Corner Singularities, Lipschitz Domains

@Techreport{MSS17_711,
  author = {F. M\"uller and D. Sch\"otzau and Ch. Schwab},
  title = {Symmetric Interior penalty discontinuous Galerkin methods for elliptic problems in polygons},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-15.pdf },
  year = {2017}
}

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