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 -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.
Mots-clés : wave propagation, finite element methods, discontinuous Galerkin methods, plane waves, ultra weak variational formulation, duality estimates, numerical dispersion
@article{M2AN_2009__43_2_297_0,
author = {Gittelson, Claude J. and Hiptmair, Ralf and Perugia, Ilaria},
title = {Plane wave discontinuous {Galerkin} methods : analysis of the $h$-version},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {297--331},
publisher = {EDP-Sciences},
volume = {43},
number = {2},
year = {2009},
doi = {10.1051/m2an/2009002},
mrnumber = {2512498},
zbl = {1165.65076},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2009002/}
}
TY - JOUR AU - Gittelson, Claude J. AU - Hiptmair, Ralf AU - Perugia, Ilaria TI - Plane wave discontinuous Galerkin methods : analysis of the $h$-version JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 297 EP - 331 VL - 43 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009002/ DO - 10.1051/m2an/2009002 LA - en ID - M2AN_2009__43_2_297_0 ER -
%0 Journal Article %A Gittelson, Claude J. %A Hiptmair, Ralf %A Perugia, Ilaria %T Plane wave discontinuous Galerkin methods : analysis of the $h$-version %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 297-331 %V 43 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009002/ %R 10.1051/m2an/2009002 %G en %F M2AN_2009__43_2_297_0
Gittelson, Claude J.; Hiptmair, Ralf; Perugia, Ilaria. Plane wave discontinuous Galerkin methods : analysis of the $h$-version. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 297-331. doi : 10.1051/m2an/2009002. http://www.numdam.org/articles/10.1051/m2an/2009002/
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