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Plane wave discontinuous Galerkin methods: Analysis of the h-version

Published online by Cambridge University Press:  07 February 2009

Claude J. Gittelson ,
Ralf Hiptmair  and
Ilaria Perugia
Claude J. Gittelson
Affiliation:
SAM, ETH Zurich, 8092 Zürich, Switzerland. giclaude@student.ethz.ch; hiptmair@sam.math.ethz.ch
Ralf Hiptmair
Affiliation:
SAM, ETH Zurich, 8092 Zürich, Switzerland. giclaude@student.ethz.ch; hiptmair@sam.math.ethz.ch
Ilaria Perugia
Affiliation:
Dipartimento di Matematica, Università di Pavia, Italy. ilaria.perugia@unipv.it
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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. We focus on 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 in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves 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 propagationfinite element methodsdiscontinuous Galerkin methodsplane wavesultra weak variational formulationduality estimatesnumerical dispersion.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 2 , March 2009 , pp. 297 - 331
Copyright
© EDP Sciences, SMAI, 2009

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