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

Handle URI:
http://hdl.handle.net/10754/597564
Title:
Analysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problems
Authors:
Kim, Seungil; Pasciak, Joseph E.
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.
Publisher:
Elsevier BV
Journal:
Journal of Mathematical Analysis and Applications
Issue Date:
Jan-2010
DOI:
10.1016/j.jmaa.2009.07.024
Type:
Article
ISSN:
0022-247X
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).
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorKim, Seungilen
dc.contributor.authorPasciak, Joseph E.en
dc.date.accessioned2016-02-25T12:42:08Zen
dc.date.available2016-02-25T12:42:08Zen
dc.date.issued2010-01en
dc.identifier.citationKim 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.en
dc.identifier.issn0022-247Xen
dc.identifier.doi10.1016/j.jmaa.2009.07.024en
dc.identifier.urihttp://hdl.handle.net/10754/597564en
dc.description.abstractIn 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.en
dc.description.sponsorshipThis 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).en
dc.publisherElsevier BVen
dc.subjectBounded holomorphic semigroupen
dc.subjectCartesian PMLen
dc.subjectEssential spectrumen
dc.subjectm-Sectorialen
dc.subjectPerfectly Matched Layeren
dc.subjectWeyl spectrumen
dc.subjectZhislin spectrumen
dc.titleAnalysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problemsen
dc.typeArticleen
dc.identifier.journalJournal of Mathematical Analysis and Applicationsen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.