An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number

Previous Article
Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation
IPI Home
This Issue
Next Article
Preface

August 2013, 7(3): 663-678. doi: 10.3934/ipi.2013.7.663

An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number

Zhiming Chen ^1, , Chao Liang ^1, and Xueshuang Xiang ^1,

1.	LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, China, China

Received September 2012 Revised February 2013 Published September 2013

Abstract
Related Papers

The anisotropic perfectly matched layer (PML) defines a continuous vector field outside a rectangle domain and performs the complex coordinate stretching along the direction of the vector field. In this paper we propose a new way of constructing the vector field which allows us to prove the exponential decay of the stretched Green function without the constraint on the thickness of the PML layer. We report numerical experiments to illustrate the competitive behavior of the proposed PML method.

Keywords: discontinuous wave number, Helmholtz equation, anisotropic PML..

Mathematics Subject Classification: Primary: 65N30; Secondary: 78A45, 35Q6.

Citation: Zhiming Chen, Chao Liang, Xueshuang Xiang. An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number. Inverse Problems and Imaging, 2013, 7 (3) : 663-678. doi: 10.3934/ipi.2013.7.663

References:

[1]	J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159.
[2]	J. H. Bramble and J. E. Pasciak, Analysis of a cartesian PML approximation to acoustic scattering problems in $\mathbb{R}^2$ and $\mathbb{R}^3$, J. Appl. Comput. Math., to appear.
[3]	Z. Chen and H. Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Nummer. Anal., 41 (2003), 799-826. doi: 10.1137/S0036142902400901.
[4]	Z. Chen and X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), 645-671.
[5]	Z. Chen and X. M. Wu, An adaptive uniaxial perfectly matched layer technique for time-Harmonic scattering problems, Numer. Math. Theory Methods Appl., 1 (2008), 113-137.
[6]	J. Chen and Z. Chen, An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comp., 77 (2008), 673-698. doi: 10.1090/S0025-5718-07-02055-8.
[7]	Z. Chen and W. Zheng, 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. doi: 10.1137/090750603.
[8]	Z. Chen, T. Cui and L. Zhang, An adaptive anisotropic perfectly matched layer method for 3D time harmonic electromagnetic scattering problems, Numer. Math., to appear.
[9]	W. C. Chew, "Waves and Fields in Inhomogeneous Media," Springer, New York, 1990. doi: 10.1109/9780470547052.
[10]	W. C. Chew and W. Weedon, A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates, Microwave Opt. Tech. Lett., 7 (1994), 599-604. doi: 10.1002/mop.4650071304.
[11]	F. Collino and P. B. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), 2061-2090. doi: 10.1137/S1064827596301406.
[12]	T. Hohage, F. Schmidt and L. Zschiedrich, Solving time-harmonic scattering problems based on the pole condition. II: Convergence of the PML method, SIAM J. Math. Anal., 35 (2003), 547-560 (electronic). doi: 10.1137/S0036141002406485.
[13]	S. Kim and J. E. Pasciak, Analysis of a cartisian PML approximation to acoustic scattering problems in $\mathbb{R}^2$, J. Math. Anal. Appl., 370 (2010), 168-186. doi: 10.1016/j.jmaa.2010.05.006.
[14]	M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations, Computing, 60 (1998), 229-241. doi: 10.1007/BF02684334.
[15]	M. Lassas and E. Somersalo, Analysis of the PML equations in general convex geometry, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1183-1207. doi: 10.1017/S0308210500001335.
[16]	K. C. Meza-Fajardo and A. S. Papageorgiou, 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.
[17]	A. F. Oskooi, L. Zhang, Y. Avniel and S. G. Johnson, The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers, Optical Express, 16 (2008), 11376-11392. doi: 10.1364/OE.16.011376.
[18]	F. L. Teixeira and W. C. Chew, Advances in the theory of perfectly matched layers, In: "Fast and Efficient Algorithms in Computational Electromagnetics" (eds W.C. Chew et al), Artech House, (2001), 283-346.
[19]	D. V. Trenev, "Spatial Scaling for the Numerical Approximation of Problems on Unbounded Domains," Thesis (Ph.D.), Texas A & M University, 2009, 112 pp.
[20]	E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations, Absorbing boundary conditions, Appl. Numer. Math., 27 (1998), 533-557. doi: 10.1016/S0168-9274(98)00026-9.
[21]	L. Zschiedrich, R. Klose, A. Schödle and F. Schmidt, 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. doi: 10.1016/j.cam.2005.03.047.

show all references

References:

[1]	J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159.
[2]	J. H. Bramble and J. E. Pasciak, Analysis of a cartesian PML approximation to acoustic scattering problems in $\mathbb{R}^2$ and $\mathbb{R}^3$, J. Appl. Comput. Math., to appear.
[3]	Z. Chen and H. Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Nummer. Anal., 41 (2003), 799-826. doi: 10.1137/S0036142902400901.
[4]	Z. Chen and X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), 645-671.
[5]	Z. Chen and X. M. Wu, An adaptive uniaxial perfectly matched layer technique for time-Harmonic scattering problems, Numer. Math. Theory Methods Appl., 1 (2008), 113-137.
[6]	J. Chen and Z. Chen, An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comp., 77 (2008), 673-698. doi: 10.1090/S0025-5718-07-02055-8.
[7]	Z. Chen and W. Zheng, 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. doi: 10.1137/090750603.
[8]	Z. Chen, T. Cui and L. Zhang, An adaptive anisotropic perfectly matched layer method for 3D time harmonic electromagnetic scattering problems, Numer. Math., to appear.
[9]	W. C. Chew, "Waves and Fields in Inhomogeneous Media," Springer, New York, 1990. doi: 10.1109/9780470547052.
[10]	W. C. Chew and W. Weedon, A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates, Microwave Opt. Tech. Lett., 7 (1994), 599-604. doi: 10.1002/mop.4650071304.
[11]	F. Collino and P. B. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), 2061-2090. doi: 10.1137/S1064827596301406.
[12]	T. Hohage, F. Schmidt and L. Zschiedrich, Solving time-harmonic scattering problems based on the pole condition. II: Convergence of the PML method, SIAM J. Math. Anal., 35 (2003), 547-560 (electronic). doi: 10.1137/S0036141002406485.
[13]	S. Kim and J. E. Pasciak, Analysis of a cartisian PML approximation to acoustic scattering problems in $\mathbb{R}^2$, J. Math. Anal. Appl., 370 (2010), 168-186. doi: 10.1016/j.jmaa.2010.05.006.
[14]	M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations, Computing, 60 (1998), 229-241. doi: 10.1007/BF02684334.
[15]	M. Lassas and E. Somersalo, Analysis of the PML equations in general convex geometry, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1183-1207. doi: 10.1017/S0308210500001335.
[16]	K. C. Meza-Fajardo and A. S. Papageorgiou, 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.
[17]	A. F. Oskooi, L. Zhang, Y. Avniel and S. G. Johnson, The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers, Optical Express, 16 (2008), 11376-11392. doi: 10.1364/OE.16.011376.
[18]	F. L. Teixeira and W. C. Chew, Advances in the theory of perfectly matched layers, In: "Fast and Efficient Algorithms in Computational Electromagnetics" (eds W.C. Chew et al), Artech House, (2001), 283-346.
[19]	D. V. Trenev, "Spatial Scaling for the Numerical Approximation of Problems on Unbounded Domains," Thesis (Ph.D.), Texas A & M University, 2009, 112 pp.
[20]	E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations, Absorbing boundary conditions, Appl. Numer. Math., 27 (1998), 533-557. doi: 10.1016/S0168-9274(98)00026-9.
[21]	L. Zschiedrich, R. Klose, A. Schödle and F. Schmidt, 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. doi: 10.1016/j.cam.2005.03.047.

[1]	S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604
[2]	Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
[3]	John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems and Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333
[4]	Sang-Yeun Shim, Marcos Capistran, Yu Chen. Rapid perturbational calculations for the Helmholtz equation in two dimensions. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 627-636. doi: 10.3934/dcds.2007.18.627
[5]	Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
[6]	Qingjie Hu, Zhihao Ge, Yinnian He. Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2923-2948. doi: 10.3934/dcdsb.2020046
[7]	Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063
[8]	Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems and Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[9]	Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[10]	Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050
[11]	Sergey Degtyarev. Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022029
[12]	Giuseppe Maria Coclite, Lorenzo di Ruvo. Discontinuous solutions for the generalized short pulse equation. Evolution Equations and Control Theory, 2019, 8 (4) : 737-753. doi: 10.3934/eect.2019036
[13]	Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems and Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
[14]	Yulong Xing, Ching-Shan Chou, Chi-Wang Shu. Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Problems and Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967
[15]	José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025
[16]	B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems and Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405
[17]	Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[18]	Karzan Berdawood, Abdeljalil Nachaoui, Rostam Saeed, Mourad Nachaoui, Fatima Aboud. An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 57-78. doi: 10.3934/dcdss.2021013
[19]	María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049
[20]	Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Zhiming Chen Chao Liang Xueshuang Xiang
on Google Scholar
Zhiming Chen Chao Liang Xueshuang Xiang

[Back to Top]