# American Institute of Mathematical Sciences

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

 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

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.
Citation: Zhiming Chen, Chao Liang, Xueshuang Xiang. An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number. Inverse Problems & 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.  doi: 10.1006/jcph.1994.1159.  Google Scholar [2] J. H. Bramble and J. E. Pasciak, Analysis of a cartesian PML approximation to acoustic scattering problems in $\mathbbR^2$ and $\mathbbR^3$,, J. Appl. Comput. Math., ().   Google Scholar [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.  doi: 10.1137/S0036142902400901.  Google Scholar [4] Z. Chen and X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems,, SIAM J. Numer. Anal., 43 (2005), 645.   Google Scholar [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.   Google Scholar [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.  doi: 10.1090/S0025-5718-07-02055-8.  Google Scholar [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.  doi: 10.1137/090750603.  Google Scholar [8] Z. Chen, T. Cui and L. Zhang, An adaptive anisotropic perfectly matched layer method for 3D time harmonic electromagnetic scattering problems,, Numer. Math., ().   Google Scholar [9] W. C. Chew, "Waves and Fields in Inhomogeneous Media,", Springer, (1990).  doi: 10.1109/9780470547052.  Google Scholar [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.  doi: 10.1002/mop.4650071304.  Google Scholar [11] F. Collino and P. B. Monk, The perfectly matched layer in curvilinear coordinates,, SIAM J. Sci. Comput., 19 (1998), 2061.  doi: 10.1137/S1064827596301406.  Google Scholar [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.  doi: 10.1137/S0036141002406485.  Google Scholar [13] S. Kim and J. E. Pasciak, Analysis of a cartisian PML approximation to acoustic scattering problems in $\mathbbR^2$,, J. Math. Anal. Appl., 370 (2010), 168.  doi: 10.1016/j.jmaa.2010.05.006.  Google Scholar [14] M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations,, Computing, 60 (1998), 229.  doi: 10.1007/BF02684334.  Google Scholar [15] M. Lassas and E. Somersalo, Analysis of the PML equations in general convex geometry,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1183.  doi: 10.1017/S0308210500001335.  Google Scholar [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.   Google Scholar [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.  doi: 10.1364/OE.16.011376.  Google Scholar [18] F. L. Teixeira and W. C. Chew, Advances in the theory of perfectly matched layers,, In:, (2001), 283.   Google Scholar [19] D. V. Trenev, "Spatial Scaling for the Numerical Approximation of Problems on Unbounded Domains,", Thesis (Ph.D.), (2009).   Google Scholar [20] E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations,, Absorbing boundary conditions, 27 (1998), 533.  doi: 10.1016/S0168-9274(98)00026-9.  Google Scholar [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.  doi: 10.1016/j.cam.2005.03.047.  Google Scholar

show all references

##### References:
 [1] J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves,, J. Comput. Phys., 114 (1994), 185.  doi: 10.1006/jcph.1994.1159.  Google Scholar [2] J. H. Bramble and J. E. Pasciak, Analysis of a cartesian PML approximation to acoustic scattering problems in $\mathbbR^2$ and $\mathbbR^3$,, J. Appl. Comput. Math., ().   Google Scholar [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.  doi: 10.1137/S0036142902400901.  Google Scholar [4] Z. Chen and X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems,, SIAM J. Numer. Anal., 43 (2005), 645.   Google Scholar [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.   Google Scholar [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.  doi: 10.1090/S0025-5718-07-02055-8.  Google Scholar [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.  doi: 10.1137/090750603.  Google Scholar [8] Z. Chen, T. Cui and L. Zhang, An adaptive anisotropic perfectly matched layer method for 3D time harmonic electromagnetic scattering problems,, Numer. Math., ().   Google Scholar [9] W. C. Chew, "Waves and Fields in Inhomogeneous Media,", Springer, (1990).  doi: 10.1109/9780470547052.  Google Scholar [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.  doi: 10.1002/mop.4650071304.  Google Scholar [11] F. Collino and P. B. Monk, The perfectly matched layer in curvilinear coordinates,, SIAM J. Sci. Comput., 19 (1998), 2061.  doi: 10.1137/S1064827596301406.  Google Scholar [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.  doi: 10.1137/S0036141002406485.  Google Scholar [13] S. Kim and J. E. Pasciak, Analysis of a cartisian PML approximation to acoustic scattering problems in $\mathbbR^2$,, J. Math. Anal. Appl., 370 (2010), 168.  doi: 10.1016/j.jmaa.2010.05.006.  Google Scholar [14] M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations,, Computing, 60 (1998), 229.  doi: 10.1007/BF02684334.  Google Scholar [15] M. Lassas and E. Somersalo, Analysis of the PML equations in general convex geometry,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1183.  doi: 10.1017/S0308210500001335.  Google Scholar [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.   Google Scholar [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.  doi: 10.1364/OE.16.011376.  Google Scholar [18] F. L. Teixeira and W. C. Chew, Advances in the theory of perfectly matched layers,, In:, (2001), 283.   Google Scholar [19] D. V. Trenev, "Spatial Scaling for the Numerical Approximation of Problems on Unbounded Domains,", Thesis (Ph.D.), (2009).   Google Scholar [20] E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations,, Absorbing boundary conditions, 27 (1998), 533.  doi: 10.1016/S0168-9274(98)00026-9.  Google Scholar [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.  doi: 10.1016/j.cam.2005.03.047.  Google Scholar
 [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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020046 [7] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [8] Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050 [9] Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063 [10] B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405 [11] Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051 [12] Giuseppe Maria Coclite, Lorenzo di Ruvo. Discontinuous solutions for the generalized short pulse equation. Evolution Equations & Control Theory, 2019, 8 (4) : 737-753. doi: 10.3934/eect.2019036 [13] Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 [14] Yulong Xing, Ching-Shan Chou, Chi-Wang Shu. Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Problems & Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967 [15] Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391 [16] V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611 [17] Eugenio Sinestrari. Wave equation with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 881-896. doi: 10.3934/dcds.1999.5.881 [18] Baptiste Fedele, Claudia Negulescu. Numerical study of an anisotropic Vlasov equation arising in plasma physics. Kinetic & Related Models, 2018, 11 (6) : 1395-1426. doi: 10.3934/krm.2018055 [19] Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297 [20] Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831

2018 Impact Factor: 1.469