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Discontinuous Galerkin methods for acoustic wave propagation in polygons

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

(Report number 2018-02)

Abstract
We analyze space semi-discretization of linear, second-order wave equation by discontinuous Galerkin methods in two-dimensional polygonal domains where solutions exhibit singular behavior near corners. To resolve these singularities, we consider two families of locally refined meshes: graded meshes and bisection refinement meshes. We prove that for appropriately chosen refinement parameters, optimal asymptotic rates of convergence with respect to the total number of degrees of freedom are obtained, both in the energy norm errors and the \(\mathcal{L}^2\)-norm errors. The theoretical convergence orders are confirmed in a series of numerical experiments which also indicate that analogous results hold for incompatible data which is not covered by the currently available regularity theory.

Keywords: Linear Wave Equations, Finite Element Methods, Discontinuous Galerkin Methods, Corner Singularities, Lipschitz Domains

@Techreport{MSS18_756,
  author = {F. M\"uller and D. Sch\"otzau and Ch. Schwab},
  title = {Discontinuous Galerkin methods for acoustic wave propagation in polygons},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-02},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-02.pdf },
  year = {2018}
}

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