Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T05:13:12.321Z Has data issue: false hasContentIssue false
  • English
  • Français

A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions

Published online by Cambridge University Press:  03 June 2015

Long Yuan  and
Qiya Hu
Long Yuan*
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Qiya Hu*
Affiliation:
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
Email: [email protected]
Corresponding author. Email: [email protected]
Get access

Abstract

An interesting discretization method for Helmholtz equations was introduced in B. Després [1]. This method is based on the ultra weak variational formulation (UWVF) and the wave shape functions, which are exact solutions of the governing Helmholtz equation. In this paper we are concerned with fast solver for the system generated by the method in [1]. We propose a new preconditioner for such system, which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in [13]. In our numerical experiments, this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations, and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.

Keywords

65F0865N55Helmholtz equationultra weak variational formulationwave shape functionspreconditioneriteration counts
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 6 , December 2013 , pp. 791 - 808
Copyright
Copyright © Global-Science Press 2013

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]Després, B., Méthod de DÉComposition de Domaine Pour Les Problèmes de Propagation D’Ondes en RÉGime Harmonique, Le théoréme de Borg Pour L’Équation de Hill Vectorielle, Ph.D.thesis, FRANCE/Paris IX Dauphine, 1991.Google Scholar
[2]Potier, C. and Martret, R., Finite volume solution of Maxwell’s equations in non-steady mode, La Recherche Aérospatiale, 5 (1994), pp. 329–342.Google Scholar
[3]Yee, K., Numerical solution of initial boundary value problem in isotropic meida, IEEE Trans. Antennas Propagation, AP-14 (1966), pp. 302–307.Google Scholar
[4]Deraemaeker, A., Babuska, I. and Bouillard, P., Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions, Int. J. Numer. Meth. Eng., 46 (1999), pp. 471–499.Google Scholar
[5]Freymann, R., Advanced Numerical and Experimental Methods in the Field of Vehicle Structural acoustics, Hieronymus Buchreproduktions GmbH, Muünchen, 2000.Google Scholar
[6]Harari, I. and Hughes, T., Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains, Comput. Meth. Appl. Mech. Eng., 98(3) (1992), pp. 411–454.CrossRefGoogle Scholar
[7]Babuska, I., Ihlenburg, F., Paik, E. and Sauter, S., A generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution, Comput. Meth. Appl. Mech. Eng., 128 (1995), pp. 325–359.CrossRefGoogle Scholar
[8]Melenk, J. and Babuska, I., The partition of unity finite element method: Basic theory and applications, Comput. Meth. Appl. Mech. Eng., 139 (1996), pp. 289–314.CrossRefGoogle Scholar
[9]Melenk, J. and Babuska, I., Approximation with harmonic and generalized harmonic polynomials in the partition of unity method, Comput. Assist. Mech. Eng. Sci., 4 (1997), pp. 607–632.Google Scholar
[10]Soize, C., Reduced models in the medium frequency range for the general dissipative structural dynamic systems, Euro. J. Mech. A/Solids, 17 (1998), pp. 657–685.CrossRefGoogle Scholar
[11]Langre, E., Fonctions de transfert de plaques par équations intégrales, Test de validation et de performance, Rapport CEA. DMT/90/395 (1991).Google Scholar
[12]Harari, I. and Hugues, T., A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics, Comput. Meth. Appl. Mech. Eng., 97 (1992) pp. 77–102.Google Scholar
[13]Cessenat, O. and Despres, B., Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional helmholtz problem, SIAM J. Numer. Anal., 35(1) (1998), pp. 255–299.Google Scholar
[14]Huttunen, T., Monk, P. and Kaipio, J., Computational aspects of the Ultra-weak variational formulation, J. Comput. Phys., 182(1) (2002), pp. 27–46.Google Scholar
[15]Riou, H., Ladevèze, P. and Sourcis, B., The multiscale VTCR approach applied to acoustics problems, J. Comput. Acous., 16 (2008), pp. 487–505.Google Scholar
[16]Huttunen, T., Malinen, M. and Monk, P., Solving Maxwell’s equations using the ultra weak variational formulation, J. Comput. Phys., 223 (2007), pp. 731–758.Google Scholar
[17]Opfer, G. and Schober, G., Richardson’s iteration for nonsymmetric matrices, Linear Algebra. Appl., 58 (1984), pp. 343–361.Google Scholar
[18]Lions, J. and Magenes, E., Problémes aux limites non homogénes et applications, 1 (1968), et 2, Dunod, Pairs.Google Scholar
[19]Tezaur, R. and Farhat, C., Three-dimensional directional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Int. J. Numer. Meth. Eng., 66 (2006), pp. 796–815.Google Scholar
[20]Huttunen, T., Gamallo, P. and Astley, R., Comparison of two wave element methods for the Helmholtz problem, Commun. Numer. Meth. Eng., 25 (2009), pp. 35–52.Google Scholar
[21]Luostari, T., Huttunen, T. and Monk, P., The ultra weak variational formulation using Bessel basis functions, Commun. Comput. Phys., 11 (2012), pp. 400–414.Google Scholar
[22]Huttunen, T., Kaipio, J. and Monk, P., The perfectly matched layer for the ultra weak varia-tional formulation of the 3D Helmholtz equation, Int. J. Numer. Meth. Eng., 61 (2004), pp. 1072–1092.Google Scholar
[23]Hu, Qiya, Tai, Xue-Cheng. And Winther, Ragnar., A saddle point approach to the computation of harmonic maps, SIAM J. Numer. Anal., 47 (2009), pp. 1500–1523.Google Scholar