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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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The computation of resonances in open systems using a perfectly matched layer
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by Seungil Kim and Joseph E. Pasciak PDF
Math. Comp. 78 (2009), 1375-1398 Request permission


In this paper, we consider the problem of computing resonances in open systems. We first characterize resonances in terms of (improper) eigenfunctions of the Helmholtz operator on an unbounded domain. The perfectly matched layer (PML) technique has been successfully applied to the computation of scattering problems. We shall see that the application of PML converts the resonance problem to a standard eigenvalue problem (still on an infinite domain). This new eigenvalue problem involves an operator which resembles the original Helmholtz equation transformed by a complex shift in the coordinate system. Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements on the computational domain. We shall prove that the first of these steps leads to eigenvalue convergence (to the desired resonance values) which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large. The analysis of the second step is classical. Finally, we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions.
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Additional Information
  • Seungil Kim
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Email:
  • Joseph E. Pasciak
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Email:
  • Received by editor(s): July 9, 2007
  • Received by editor(s) in revised form: July 22, 2008
  • Published electronically: February 6, 2009
  • Additional Notes: This work was supported in part by the National Science Foundation through grant DMS-0609544.
  • © Copyright 2009 American Mathematical Society
  • Journal: Math. Comp. 78 (2009), 1375-1398
  • MSC (2000): Primary 65N30, 78M10
  • DOI:
  • MathSciNet review: 2501055