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**Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems.**
*(English)*
Zbl 1116.78019

Authors’ summary: We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of M. Lassas and E. Somersalo [Computing 60, No. 3, 229–241 (1998; Zbl 0899.35026)], a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius \( R_t\). We also show exponential (in the parameter \( R_t\)) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer. Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

Reviewer: Aleksander Pankov (Baltimore)

### MSC:

78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |

65F10 | Iterative numerical methods for linear systems |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

### Keywords:

Maxwell’s equations; Helmholtz equation; time-harmonic acoustic and electromagnetic scattering; div-curl systems; perfectly matched layer; PML### Citations:

Zbl 0899.35026
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\textit{J. H. Bramble} and \textit{J. E. Pasciak}, Math. Comput. 76, No. 258, 597--614 (2007; Zbl 1116.78019)

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### References:

[1] | C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823 – 864 (English, with English and French summaries). , https://doi.org/10.1002/(SICI)1099-1476(199806)21:93.0.CO;2-B · Zbl 0914.35094 |

[2] | Jean-Pierre Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), no. 2, 185 – 200. · Zbl 0814.65129 |

[3] | Jean-Pierre Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 127 (1996), no. 2, 363 – 379. · Zbl 0862.65080 |

[4] | W. Chew and W. Weedon. A 3d perfectly matched medium for modified Maxwell’s equations with streched coordinates. Microwave Opt. Techno. Lett., 13(7):599-604, 1994. |

[5] | Francis Collino and Peter Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput. 19 (1998), no. 6, 2061 – 2090. · Zbl 0940.78011 |

[6] | V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. · Zbl 0413.65081 |

[7] | C. I. Goldstein, The finite element method with nonuniform mesh sizes applied to the exterior Helmholtz problem, Numer. Math. 38 (1981/82), no. 1, 61 – 82. · Zbl 0445.65102 |

[8] | Marcus J. Grote and Joseph B. Keller, Nonreflecting boundary conditions for Maxwell’s equations, J. Comput. Phys. 139 (1998), no. 2, 327 – 342. · Zbl 0908.65118 |

[9] | N. Kantartzis, P. Petropoulis, and T. Tsiboukis. A comparison of the Grote-Keller exact ABC and the well posed PML for Maxwell’s equations in spherical coordinates. IEEE Trans. on Magnetics, 35:1418-1422, 1999. |

[10] | M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations, Computing 60 (1998), no. 3, 229 – 241. · Zbl 0899.35026 |

[11] | Matti Lassas and Erkki Somersalo, Analysis of the PML equations in general convex geometry, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 5, 1183 – 1207. · Zbl 1200.35013 |

[12] | Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. · Zbl 1024.78009 |

[13] | Jaak Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 279 – 317 (French). · Zbl 0151.17903 |

[14] | Peter G. Petropoulos, Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates, SIAM J. Appl. Math. 60 (2000), no. 3, 1037 – 1058. · Zbl 1025.78016 |

[15] | Luc Tartar, Topics in nonlinear analysis, Publications Mathématiques d’Orsay 78, vol. 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978. · Zbl 0395.00008 |

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