Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T12:29:48.788Z Has data issue: false hasContentIssue false
  • English
  • Français

A Modified Discontinuous Galerkin Method for Solving Efficiently Helmholtz Problems

Published online by Cambridge University Press:  20 August 2015

Magdalena Grigoroscuta-Strugaru ,
Mohamed Amara ,
Henri Calandra  and
Rabia Djellouli
Magdalena Grigoroscuta-Strugaru*
Affiliation:
INRIA Bordeaux Sud-Ouest Research Center, Team Project Magique-3D and LMA/CNRS UMR 5142, Université de Pau et des Pays de l’Adour, France BCAM, Basque Center for Applied Mathematics, Bilbao, Spain
Mohamed Amara*
Affiliation:
INRIA Bordeaux Sud-Ouest Research Center, Team Project Magique-3D and LMA/CNRS UMR 5142, Université de Pau et des Pays de l’Adour, France
Henri Calandra*
Affiliation:
TOTAL, Avenue Larribau, Pau, France
Rabia Djellouli*
Affiliation:
Department of Mathematics, California State University Northridge and Interdisciplinary Research Institute for the Sciences, IRIS, USA
*
Email address:[email protected]
Email address:[email protected]
Email address:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

A new solution methodology is proposed for solving efficiently Helmholtz problems. The proposed method falls in the category of the discontinuous Galerkin methods. However, unlike the existing solution methodologies, this method requires solving (a) well-posed local problems to determine the primal variable, and (b) a global positive semi-definite Hermitian system to evaluate the Lagrange multiplier needed to restore the continuity across the element edges. Illustrative numerical results obtained for two-dimensional interior Helmholtz problems are presented to assess the accuracy and the stability of the proposed solution methodology.

Keywords

35J0565N3049M2735L0593E24Helmholtz equationdiscontinuous Galerkinplane wavesLagrange multipliersinf-sup conditionwaveguide problems
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 335 - 350
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Amara, M., Barucq, H., Bernardini, A., Djellouli, R., On the numerical performance of a new discretization scheme for solving Helmholtz problems, Theoretical and Computational Acoustics, Taroudakis, M. and Papadakis, P. eds. (2008), 139–146.Google Scholar
[2]Amara, M., Djellouli, R., Farhat, C., Convergence analysis of a discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of Helmholtz problems, SIAM J. Numer. Anal., 47 (2009), 1038–1066.CrossRefGoogle Scholar
[3]Amara, M., Calandra, H., Djellouli, R., Grigoroscuta-Strugaru, M., A modified discontinuous Galerkin method for solving Helmholtz problems, Rapport de Recherche, INRIA, 7050, 2009, available online at http://hal.inria.fr/inria-00421584/fr/.Google Scholar
[4]Babuška, I.ˇka, Melenk, I. J. M., The partition of unity method, Int. J. Numer. Meth. Eng., 40 (1997), 727–758.Google Scholar
[5]Babuška, I., Sauter, S., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., (34) 1997, 2392–2423.Google Scholar
[6]Desmet, W., Hal, B. van, Sas, P., Vandepitte, D., A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems, Adv. Engr. Softw., 33 (2002), 527–540.CrossRefGoogle Scholar
[7]Cessenat, O., Despres, B., Application of an ultra-weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problems, SIAM J. Numer. Anal., 35 (1998), 255–299.Google Scholar
[8]Farhat, C., Harari, I., Hetmaniuk, U., A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Engr. 192 (2003), 1389–1419.Google Scholar
[9]Farhat, C., Wiedemann-Goiran, P., Tezaur, R., A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes, Wave Motion, 39 (2004), 307–317.CrossRefGoogle Scholar
[10]Farhat, C., Tezaur, R., Wiedemann-Goiran, P., Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems, Int. J. Numer. Meth. Eng., 61 (2004), 1938–1956.CrossRefGoogle Scholar
[11]Franca, L.P., Farhat, C., Macedo, A.P., Lesoinne, M., Residual-free bubbles for the Helmholtz equation, Int. J. Numer. Meth. Eng., 40 (1997), 4003–4009.Google Scholar
[12]Genechten, B. V., Bergen, B., Vandepitte, D., Desmet, W., A Trefftz-based numerical modeling framework for Helmholtz problems with complex multiple-scatterer configurations, J. Comput. Phys., 229 (2010), 6623–6643.CrossRefGoogle Scholar
[13]Grigoroscuta-Strugaru, M., Contribution à la résolution numérique des problemes de Helmholtz, University of Pau, France, December 2009, available online at http://tel.archives-ouvertes.fr/index.php?halsid=61bpnng9qev5snpremhtjku7o7&view_this_doc=tel-00473486&version=2Google Scholar
[14]Grisvard, P., Elliptic Problems in Nonsmooth Domains Pitman, Boston, 1985Google Scholar
[15]Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press, New Haven, 1923.Google Scholar
[16]Hörmander, L., The Analysis of Linear Partial Differential Operator, Springer-Verlag, New York, 1985.Google Scholar
[17]Ihlenburg, F., Finite Element Analysis of Acoustic Scattering, Appl. Math. Sci., 132, Springer-Verlag, New York, 1998.Google Scholar
[18]Magoules, F., Computational Methods for Acoustics Problems, Saxe-Coburg Publications, 2008.Google Scholar
[19]Monk, P., Wang, D. Q., A least-squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 175 (1999), 121–136.Google Scholar
[20]Rose, M. E., Weak element approximations to elliptic differential equations, Numer. Math., 24 (1975), 185–204.Google Scholar
[21]Schenk, O., Gärtner, K., Solving unsymmetric sparse systems of linear equations with PAR-DISO, J. Future Generation Comp. Sys., 20 (2004), 475–487.Google Scholar
[22]Schenk, O., Gärtner, K., On fast factorization pivoting methods for symmetric indefinite systems, Elec. Trans. Numer. Anal., 23 (2006), 158–179.Google Scholar
[23]Stojek, M., Least-squares Trefftz-type elements for the Helmholtz equation, Int. J. Numer. Meth. Eng., 41 (1998), 831–849.Google Scholar
[24]Taylor, M. E., Partial Differential Equations I: Basic Theory, Springer-Verlag, New York 1997.Google Scholar