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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]
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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

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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