# 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

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##### 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
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