Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T15:23:36.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of an Anisotropic Perfectly Matched Layer Method for Helmholtz Scattering Problems

Part of: Basic linear algebra Numerical linear algebra

Published online by Cambridge University Press:  20 July 2016

Chao Liang  and
Xueshuang Xiang
Chao Liang*
Affiliation:
Science and Technology on Electronic Information Control Laboratory, Chengdu 610036, China LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Xueshuang Xiang*
Affiliation:
Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology, Beijing 100094, China
*
*Corresponding author. Email addresses:liangchao@lsec.cc.ac.cn (C. Liang), xiangxueshuang@qxslab.cn (X. S. Xiang)
*Corresponding author. Email addresses:liangchao@lsec.cc.ac.cn (C. Liang), xiangxueshuang@qxslab.cn (X. S. Xiang)
Get access

Abstract

The anisotropic perfectly matched layer (APML) defines a continuous vector field outside a rectangle domain and performs the complex coordinate stretching along the vector field. Inspired by [Z. Chen et al., Inverse Probl. Imag., 7, (2013):663–678] and based on the idea of the shortest distance, we propose a new approach to construct the vector field which still allows us to prove the exponential decay of the stretched Green function without the constraint on the thickness of the PML layer. Moreover, by using the reflection argument, we prove the stability of the PML problem in the PML layer and the convergence of the PML method. Numerical experiments are also included.

Keywords

Anisotropic PMLHelmholtz equationReflection argumentExponential convergence

MSC classification

Secondary: 15A12: Conditioning of matrices 65F10: Iterative methods for linear systems 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 3 , August 2016 , pp. 358 - 382
Copyright
Copyright © Global-Science Press 2016 

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]Bérenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), 185–200.Google Scholar
[2]Bramble, J. H. and Pasciak, J. E., Analysis of a Cartesian PML approximation to acoustic scattering problems in ℝ2 and ℝ3, J. Comput. Appl. Math. 247 (2013), 209–230.Google Scholar
[3]Chen, J. and Chen, Z., An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comp. 77 (2008), 673–698.CrossRefGoogle Scholar
[4]Chen, Z., Cui, T. and Zhang, L., An adaptive anisotropic perfectly matched layer method for 3D time harmonic electromagnetic scattering problems, Numer. Math. 125 (2013), 639–677.CrossRefGoogle Scholar
[5]Chen, Z., Liang, C. and Xiang, X.An anisotropic perfectly matched layer method for Helmholtz scattering problem with discontinuous wave number, Inverse Probl. Imag. 7 (2013), 663–678.CrossRefGoogle Scholar
[6]Chen, Z. and Liu, X., An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal. 43 (2005), 645–671.CrossRefGoogle Scholar
[7]Chen, Z. and Wu, H., An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal. 41 (2003), 799–826.CrossRefGoogle Scholar
[8]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), 113–137.Google Scholar
[9]Chen, Z. and Xiang, X.A source transfer domain decomposition method for Helmholtz equations in unbounded domain, SIAM J. Numer. Anal. 51 (2013), 2331–2356.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 (2010), 2158–2185.CrossRefGoogle Scholar
[11]Chew, W. C., Waves and Fields in Inhomogeneous Media, Springer, New York, 1990.Google Scholar
[12]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), 599–604.CrossRefGoogle Scholar
[13]Collino, F. and Monk, P. B., The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput. 19 (1998), 2061–2090.Google Scholar
[14]Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.Google Scholar
[15]Hohage, T., Schmidt, F., and Zschiedrich, L., Solving time-harmonic scattering problems based on the pole condition. II: Convergence of the PML method, SIAM J. Math. Anal. 35 (2003), 547–560.CrossRefGoogle Scholar
[16]Kim, S. and Pasciak, J. E., Analysis of a Cartisian PML approximation to acoustic scattering problems in ℝ2, J. Math. Anal. Appl. 370 (2010), 168–186.Google Scholar
[17]Lassas, M. and Somersalo, E., On the existence and convergence of the solution of PML equations, Computing. 60 (1998), 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), 1183–1207.CrossRefGoogle Scholar
[19]Meza-Fajardo, K. C. and Papageorgiou, A. S., A nonconventional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: stability analysis, Bulletin Seismological Soc. Am. 98 (2008), 1811–1836.CrossRefGoogle Scholar
[20]McLean, W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Universtiy Press, Cambridge, 2000.Google Scholar
[21]Oskooi, A. F., Zhang, L., Avniel, Y., and Johnson, S. G., The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers, Optical Express. 16 (2008), 11376–11392.CrossRefGoogle ScholarPubMed
[22]Teixeira, F. L. and Chew, W. C., Advances in the theory of perfectly matched layers, In: “Fast and Efficient Algorithms in Computational Electromagnetics” (eds Chew, W. C.et al.), Artech House, (2001), 283–346.Google Scholar
[23]Trenev, D. V., Spatial Scaling for the Numerical Approximation of Problems on Unbounded Domains, PhD Thesis, Texas A & M University, 2009.Google Scholar
[24]Turkel, E. and Yefet, A., Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math. 27 (1998), 533–557.Google Scholar
[25]Zschiedrich, L., Klose, R., Schödle, A., and Schmidt, F., A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in two dimensions, J. Comput. Appl. Math. 188 (2006), 12–32.CrossRefGoogle Scholar