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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/m2an:2008033/
[1] , and , Dispersive and dissipative properties of discontinuous Galerkin methods for the wave equation. J. Sci. Comput. 27 (2006) 5-40. | MR | Zbl
[2] , , and , Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779. | MR | Zbl
[3] and , Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comp. Phys. (to appear). | MR
[4] , A GSVD formulation of a domain decomposition method for planar eigenvalue problems. IMA J. Numer. Anal. 27 (2007) 451-478. | MR
[5] , Application d'une nouvelle formulation variationnelle aux équations d'ondes harmoniques. Problèmes de Helmholtz 2D et de Maxwell 3D. Ph.D. thesis, Université Paris IX Dauphine, France (1996).
[6] and , Application of the ultra-weak variational formulation of elliptic PDEs to the 2-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35 (1998) 255-299. | MR | Zbl
[7] and , Using plane waves as base functions for solving time harmonic equations with the Ultra Weak Variational Formulation. J. Comput. Acoustics 11 (2003) 227-238. | MR
[8] and , Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math. Models Methods Appl. Sci. 16 (2006) 139-160. | MR | Zbl
[9] and , A comparison of two Trefftz-type methods: The ultraweak variational formulation and the least-squares method, for solving shortwave 2-D Helmholtz problems. Int. J. Numer. Meth. Eng. 71 (2007) 406-432. | MR
[10] , and , Plane wave discontinuous Galerkin methods. Preprint NI07088-HOP, Isaac Newton Institute Cambride, Cambridge, UK, December (2007) http://www.newton.cam.ac.uk/preprints/NI07088.pdf.
[11] , Boundary Methods: an Algebraic Theory. Pitman (1984). | MR | Zbl
[12] , and , The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation. Int. J. Numer. Meth. Eng. 61 (2004) 1072-1092. | MR | Zbl
[13] , and , Computational aspects of the Ultra Weak Variational Formulation. J. Comput. Phys. 182 (2002) 27-46. | MR | Zbl
[14] , , and , The Ultra Weak Variational Formulation for elastic wave problems. SIAM J. Sci. Comput. 25 (2004) 1717-1742. | MR | Zbl
[15] , and , Solving Maxwell's equations using the Ultra Weak Variational Formulation. J. Comput. Phys. 223 (2007) 731-758. | MR | Zbl
[16] , On generalized finite element methods. Ph.D. thesis, University of Maryland, College Park, USA (1995).
[17] and , The partition of unity finite element method: Basic theory and applications. Comput. Meth. Appl. Mech. Eng. 139 (1996) 289-314. | MR | Zbl
[18] and , A least squares method for the Helmholtz equation. Comput. Meth. Appl. Mech. Eng. 175 (1999) 121-136. | MR | Zbl
[19] , Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Numer. Meth. Eng. 41 (1998) 831-849. | MR | Zbl
[20] and , Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Meth. Eng. 66 (2006) 796-815. | MR | Zbl
[21] , Ein gegenstück zum Ritz'schen verfahren, in Proc. 2nd Int. Congr. Appl. Mech., Zurich (1926) 131-137. | JFM
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