Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Annalisa Buffa; Peter Monk

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 6, page 925-940
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Buffa, Annalisa, and Monk, Peter. "Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation." ESAIM: Mathematical Modelling and Numerical Analysis 42.6 (2008): 925-940. <http://eudml.org/doc/250381>.

@article{Buffa2008,
abstract = { 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. },
author = {Buffa, Annalisa, Monk, Peter},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Helmholtz equation; UWVF; plane waves; error estimate.; error estimate; ultra weak variational formulation; convergence; discontinuous Galerkin method; numerical results; Maxwell's equations; elasticity},
language = {eng},
month = {8},
number = {6},
pages = {925-940},
publisher = {EDP Sciences},
title = {Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation},
url = {http://eudml.org/doc/250381},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Buffa, Annalisa
AU - Monk, Peter
TI - Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/8//
PB - EDP Sciences
VL - 42
IS - 6
SP - 925
EP - 940
AB - 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.
LA - eng
KW - Helmholtz equation; UWVF; plane waves; error estimate.; error estimate; ultra weak variational formulation; convergence; discontinuous Galerkin method; numerical results; Maxwell's equations; elasticity
UR - http://eudml.org/doc/250381
ER -

References

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Citations in EuDML Documents

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  1. Christian Wieners, Barbara Wohlmuth, Robust operator estimates and the application to substructuring methods for first-order systems
  2. Teemu Luostari, Tomi Huttunen, Peter Monk, Error estimates for the ultra weak variational formulation in linear elasticity
  3. Teemu Luostari, Tomi Huttunen, Peter Monk, Error estimates for the ultra weak variational formulation in linear elasticity
  4. Claude J. Gittelson, Ralf Hiptmair, Ilaria Perugia, Plane wave discontinuous Galerkin methods: Analysis of the -version

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