Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-29T15:21:13.088Z Has data issue: false hasContentIssue false
  • English
  • Français

A Source Transfer Domain Decomposition Method For Helmholtz Equations in Unbounded Domain Part II: Extensions

Published online by Cambridge University Press:  28 May 2015

Zhiming Chen  and
Xueshuang Xiang
Zhiming Chen*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Xueshuang Xiang*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

In this paper we extend the source transfer domain decomposition method (STDDM) introduced by the authors to solve the Helmholtz problems in two-layered media, the Helmholtz scattering problems with bounded scatterer, and Helmholtz problems in 3D unbounded domains. The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The details of STDDM is given for each extension. Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper.

Keywords

65F0865N5565Y20Helmholtz equationhigh frequency wavesPMLsource transfer
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 3 , August 2013 , pp. 538 - 555
Copyright
Copyright © Global Science Press Limited 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] Amestoy, P. R., Duff, I. S., Koster, J. and L’Excellent, J.-Y., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM Journal of Matrix Analysis and Applications, (2001), pp. 15–41.CrossRefGoogle Scholar
[2] Amestoy, P. R., Guermouche, A., L’Excellent, J.-Y. and Pralet, S., Hybrid scheduling for the parallel solution of linear systems, Parallel Computing, (2006), pp. 136–156.Google Scholar
[3] Bramble, J. H. and Pasciak, J. E., Analysis of a Cartesian PML approximation to acoustic scattering problems in ∝2 and ∝3 , Inter. J. Numer. Anal. Model., (2012), to appear.Google Scholar
[4] Benamou, J.-D. and Despres, B., A domain decomposition method for the Helmholtz equation and related optimal control problems, J. Comput. Phys., 136 (1997), pp. 68–82.CrossRefGoogle Scholar
[5] Bérenger, J.P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), pp. 185–200.CrossRefGoogle Scholar
[6] Brandt, A. and Livshits, I., Wave-ray multigrid method for standing wave equations, Electronic Trans. Numer. Anal., 6 (1997), pp. 162–181.Google Scholar
[7] Chew, W. C. and Weedon, W., A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microwave Opt. Tech. Lett., 7 (1994), pp. 599–604.CrossRefGoogle Scholar
[8] Chen, Z. and Liu, X., An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 41 (2003), pp. 799–826.CrossRefGoogle Scholar
[9] Chen, Z. and Wu, X. M., An adaptive uniaxial perfectly matched layer technique for Time-Harmonic Scattering Problems, Numerical Mathematics: Theory, Methods and Applications, 1 (2008), pp. 113–137.Google Scholar
[10] Chen, Z. and Zheng, W., Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media, SIAM J. Numer. Anal., 48 (2011), pp. 2158–2185.CrossRefGoogle Scholar
[11] Chen, Z. and Xiang, X., A source transfer domain decomposition method for Helmholtz equations in unbounded domain, submitted.Google Scholar
[12] Elman, H. C., Ernst, O. G., and O’Leary, D.P., A multigrid nethod enhanced by Krylov subspace iteration f or discrete Helmholtz equations, SIAM J. Sci. Comput., 23 (2001), pp. 1291–1315.Google Scholar
[13] Engquist, B. and Ying, L., Sweepingpreconditioner for the Helmholtz equation: Moving perfectly matched layers, Multiscle Model. Simul., 9 (2011), pp. 686–710.Google Scholar
[14] Erlangga, Y. A., Advances in iterative methods and preconditionersfor the Helmholtz equation, Arch. Comput. Methods Eng., 15 (2008), pp. 37–66.Google Scholar
[15] Gander, M. J., Magoules, F., and Nataf, F., Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., (2002), pp. 38–60.Google Scholar
[16] Kim, S. and Pasciak, J. E., Analysis of a Cartisian PML approximation to acoustic scattering problems in M2 , J. Math. Anal. Appl., 370 (2010), pp. 168–186.Google Scholar
[17] Lassas, M. and Somersalo, E., On the existence and convergence of the solution of PML equations. Computing, 60 (1998), pp. 229–241.Google Scholar
[18] Lassas, M. and Somersalo, E., Analysis of the PML equations in general convex geometry, Proc. Roy. Soc. Eding., 131 (2001), pp. 1183–1207.Google Scholar
[19] Osei-Kuffuor, D. and Saad, Y, Preconditioning Helmholtz linear systems, Technical Report, umsi-2009-30, Minnesota Supercomputer Institute, University of Minnesota, 2009.Google Scholar
[20] Teixeira, F. L. and Chew, W. C., Advances in the theory of perfectly matched layers, Fast and Efficient Algorithms in Computational Electromagnetics, (2001), pp. 283–346.Google Scholar