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Approximation properties of plane wave spaces and application to the analysis of the plane wave discontinuous Galerkin method

by A. Moiola

(Report number 2009-06)

Abstract
The main tool used is Vekua's theory for elliptic PDE with analytic coefficients. This leads to a general approximation result for solutions of the Helmholtz equation, using a finite dimensional space of plane wave functions, with respect to weighted Sobolev norms. As a consequence of these estimates, two new a priori error estimates with respect to the energy and $L^2$ norms are proved for the plane wave discontinuous Galerkin method for the homogeneous Helmholtz equation. These estimates are sharp with respect to the order of convergence in the meshsize $h$. In all the bounds, the dependence of the constants on the wavenumber is made explicit. So it is possible to assess the pollution effect.

Keywords:

@Techreport{M09_152,
  author = {A. Moiola},
  title = {Approximation properties of plane wave spaces and application to the analysis of the plane wave discontinuous Galerkin method},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-06.pdf },
  year = {2009}
}

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