## Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

HTML articles powered by AMS MathViewer

- by James H. Bramble and Joseph E. Pasciak PDF
- Math. Comp.
**76**(2007), 597-614 Request permission

## Abstract:

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 Lassas and Somersalo in 1998, 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.## References

- 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). MR**1626990**, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B - Jean-Pierre Berenger,
*A perfectly matched layer for the absorption of electromagnetic waves*, J. Comput. Phys.**114**(1994), no. 2, 185–200. MR**1294924**, DOI 10.1006/jcph.1994.1159 - Jean-Pierre Berenger,
*Three-dimensional perfectly matched layer for the absorption of electromagnetic waves*, J. Comput. Phys.**127**(1996), no. 2, 363–379. MR**1412240**, DOI 10.1006/jcph.1996.0181 - 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. - Francis Collino and Peter Monk,
*The perfectly matched layer in curvilinear coordinates*, SIAM J. Sci. Comput.**19**(1998), no. 6, 2061–2090. MR**1638033**, DOI 10.1137/S1064827596301406 - 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. MR**548867** - 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. MR**634753**, DOI 10.1007/BF01395809 - Marcus J. Grote and Joseph B. Keller,
*Nonreflecting boundary conditions for Maxwell’s equations*, J. Comput. Phys.**139**(1998), no. 2, 327–342. MR**1614098**, DOI 10.1006/jcph.1997.5881 - 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. - M. Lassas and E. Somersalo,
*On the existence and convergence of the solution of PML equations*, Computing**60**(1998), no. 3, 229–241. MR**1621305**, DOI 10.1007/BF02684334 - 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. MR**1862449**, DOI 10.1017/S0308210500001335 - Peter Monk,
*Finite element methods for Maxwell’s equations*, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR**2059447**, DOI 10.1093/acprof:oso/9780198508885.001.0001 - Jaak Peetre,
*Espaces d’interpolation et théorème de Soboleff*, Ann. Inst. Fourier (Grenoble)**16**(1966), no. fasc. 1, 279–317 (French). MR**221282** - 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. MR**1750090**, DOI 10.1137/S0036139998334688 - Luc Tartar,
*Topics in nonlinear analysis*, Publications Mathématiques d’Orsay 78, vol. 13, Université de Paris-Sud, Département de Mathématiques, Orsay, 1978. MR**532371**

## Additional Information

**James H. Bramble**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: bramble@math.tamu.edu
**Joseph E. Pasciak**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: pasciak@math.tamu.edu
- Received by editor(s): August 9, 2005
- Received by editor(s) in revised form: April 20, 2006
- Published electronically: December 7, 2006
- Additional Notes: This work was supported in part by the National Science Foundation through grant No. 0311902.
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 597-614 - MSC (2000): Primary 78M10, 65F10, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-06-01930-2
- MathSciNet review: 2291829