The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides a variational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.
Mots-clés : Helmholtz equation, UWVF, plane waves, error estimate
@article{M2AN_2008__42_6_925_0,
author = {Buffa, Annalisa and Monk, Peter},
title = {Error estimates for the ultra weak variational formulation of the {Helmholtz} equation},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {925--940},
publisher = {EDP-Sciences},
volume = {42},
number = {6},
year = {2008},
doi = {10.1051/m2an:2008033},
mrnumber = {2473314},
zbl = {1155.65094},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2008033/}
}
TY - JOUR AU - Buffa, Annalisa AU - Monk, Peter TI - Error estimates for the ultra weak variational formulation of the Helmholtz equation JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 925 EP - 940 VL - 42 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008033/ DO - 10.1051/m2an:2008033 LA - en ID - M2AN_2008__42_6_925_0 ER -
%0 Journal Article %A Buffa, Annalisa %A Monk, Peter %T Error estimates for the ultra weak variational formulation of the Helmholtz equation %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 925-940 %V 42 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008033/ %R 10.1051/m2an:2008033 %G en %F M2AN_2008__42_6_925_0
Buffa, Annalisa; Monk, Peter. Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 6, pp. 925-940. doi : 10.1051/m2an:2008033. http://www.numdam.org/articles/10.1051/m2an:2008033/
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