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

Authors: James H. Bramble and Joseph E. Pasciak
Journal: Math. Comp. 76 (2007), 597-614
MSC (2000): Primary 78M10, 65F10, 65N30
Published electronically: December 7, 2006
MathSciNet review: 2291829
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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.

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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Maxwell's equations, Helmholtz equation, time-harmonic acoustic and electromagnetic scattering, div-curl systems, perfectly matched layer, PML
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.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.