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Dispersion Analysis of Plane Wave Discontinuous Galerkin Methods

by C. Gittelson and R. Hiptmair

(Report number 2012-42)

Abstract
The plane wave discontinuous Galerkin (PWDG) method for the Helmholtz equation was introduced and analyzed in \([\)Gittelson, C., Hiptmair, R., and Perugia, I., Plane wave discontinuous Galerkin methods: Analysis of the \(h\)-version. Math. Model. Numer. Anal. 43 (2009), 297-331\(]\) as a generalization of the so-called ultra-weak variational formulation, see \([\) O.~Cessenat and B.~Despres, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255-299\(]\). The method relies on Trefftz-type local trial spaces spanned by plane waves of different directions, and links cells of the mesh through numerical fluxes in the spirit of discontinuous Galerkin methods. We conduct a partly empirical dispersion analysis of the method in a discrete translation invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves travelling in the same direction. We find agreement of the wave numbers for directions represented in the local trial spaces. For other directions the PWDG methods turn out to incur both phase and amplitude errors. This manifests itself as pollution effect haunting the \(h\)-version of the method. Our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wavenumber and number of plane waves.

Keywords: Helmholtz equation, plane wave, discontinuous Galerkin, numerical dispersion

@Techreport{GH12_495,
  author = {C. Gittelson and R. Hiptmair},
  title = {Dispersion Analysis of Plane Wave Discontinuous Galerkin Methods},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-42},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-42.pdf },
  year = {2012}
}

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