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    Analysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problems

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    Type
    Article
    Authors
    Kim, Seungil
    Pasciak, Joseph E.
    Date
    2010-01
    Permanent link to this record
    http://hdl.handle.net/10754/597564
    
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    Abstract
    In this paper, we study the spectrum of the operator which results when the Perfectly Matched Layer (PML) is applied in Cartesian geometry to the Laplacian on an unbounded domain. This is often thought of as a complex change of variables or "complex stretching." The reason that such an operator is of interest is that it can be used to provide a very effective domain truncation approach for approximating acoustic scattering problems posed on unbounded domains. Stretching associated with polar or spherical geometry lead to constant coefficient operators outside of a bounded transition layer and so even though they are on unbounded domains, they (and their numerical approximations) can be analyzed by more standard compact perturbation arguments. In contrast, operators associated with Cartesian stretching are non-constant in unbounded regions and hence cannot be analyzed via a compact perturbation approach. Alternatively, to show that the scattering problem PML operator associated with Cartesian geometry is stable for real nonzero wave numbers, we show that the essential spectrum of the higher order part only intersects the real axis at the origin. This enables us to conclude stability of the PML scattering problem from a uniqueness result given in a subsequent publication. © 2009 Elsevier Inc. All rights reserved.
    Citation
    Kim S, Pasciak JE (2010) Analysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problems. Journal of Mathematical Analysis and Applications 361: 420–430. Available: http://dx.doi.org/10.1016/j.jmaa.2009.07.024.
    Sponsors
    This work was supported in part by the National Science Foundation through Grant DMS-0609544 and in part by award number KUS-C1-016-04 made by King Abdulla University of Science and Technology (KAUST).
    Publisher
    Elsevier BV
    Journal
    Journal of Mathematical Analysis and Applications
    DOI
    10.1016/j.jmaa.2009.07.024
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.jmaa.2009.07.024
    Scopus Count
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