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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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