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Plane wave discontinuous Galerkin methods: analysis of the $h$-version

by C. J. Gittelson and R. Hiptmair and I. Perugia

(Report number 2008-04)

Abstract
We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. Among them the ultra weak variational formulation (UWVF) of Cessenat and Despres [O. CESSENAT AND B. DESPRES, Application of an ultra weak veriational formulation of elliptic PDEs to the twodimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp.255-299]. We identify the UWVF as representative of a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the h-version of these plane wave discontinuous Galerkin methods. To that end, we develop new inverse and approximation estimates for plane waves in two dimensions and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion.

Keywords: Wave propagation, finite element methods, discontinuous Galerkin methods, plane waves, ultra weak variational formulation, duality estimates, numerical dispersion

@Techreport{GHP08_34,
  author = {C. J. Gittelson and R. Hiptmair and I. Perugia},
  title = {Plane wave discontinuous Galerkin methods: analysis of the $h$-version},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-04.pdf },
  year = {2008}
}

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