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.
Original language | English (US) |
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Pages (from-to) | 420-430 |
Number of pages | 11 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 361 |
Issue number | 2 |
DOIs | |
State | Published - Jan 2010 |
Externally published | Yes |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledgements: 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).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.