Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem

Handle URI:
http://hdl.handle.net/10754/597549
Title:
Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem
Authors:
Bramble, James H.; Pasciak, Joseph E.; Trenev, Dimitar
Abstract:
We consider the application of a perfectly matched layer (PML) technique to approximate solutions to the elastic wave scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift in spherical coordinates which leads to a variable complex coefficient equation for the displacement vector posed on an infinite domain (the complement of the scatterer). The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). We prove existence and uniqueness of the solutions to the infinite domain and truncated domain PML equations (provided that the truncated domain is sufficiently large). We also show exponential convergence of the solution of the truncated PML problem to the solution of the original scattering problem in the region of interest. We then analyze a Galerkin numerical approximation to the truncated PML problem and prove that it is well posed provided that the PML damping parameter and mesh size are small enough. Finally, computational results illustrating the efficiency of the finite element PML approximation are presented. © 2010 American Mathematical Society.
Citation:
Bramble JH, Pasciak JE, Trenev D (2010) Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem. Math Comp 79: 2079–2079. Available: http://dx.doi.org/10.1090/s0025-5718-10-02355-0.
Publisher:
American Mathematical Society (AMS)
Journal:
Mathematics of Computation
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
2010
DOI:
10.1090/s0025-5718-10-02355-0
Type:
Article
ISSN:
0025-5718
Sponsors:
This work was supported in part by the National Science Foundation throughGrant DMS-0609544 and in part by award number KUS-C1-016-04 made by KingAbdulla University of Science and Technology (KAUST).
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Full metadata record

DC FieldValue Language
dc.contributor.authorBramble, James H.en
dc.contributor.authorPasciak, Joseph E.en
dc.contributor.authorTrenev, Dimitaren
dc.date.accessioned2016-02-25T12:41:51Zen
dc.date.available2016-02-25T12:41:51Zen
dc.date.issued2010en
dc.identifier.citationBramble JH, Pasciak JE, Trenev D (2010) Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem. Math Comp 79: 2079–2079. Available: http://dx.doi.org/10.1090/s0025-5718-10-02355-0.en
dc.identifier.issn0025-5718en
dc.identifier.doi10.1090/s0025-5718-10-02355-0en
dc.identifier.urihttp://hdl.handle.net/10754/597549en
dc.description.abstractWe consider the application of a perfectly matched layer (PML) technique to approximate solutions to the elastic wave scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift in spherical coordinates which leads to a variable complex coefficient equation for the displacement vector posed on an infinite domain (the complement of the scatterer). The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). We prove existence and uniqueness of the solutions to the infinite domain and truncated domain PML equations (provided that the truncated domain is sufficiently large). We also show exponential convergence of the solution of the truncated PML problem to the solution of the original scattering problem in the region of interest. We then analyze a Galerkin numerical approximation to the truncated PML problem and prove that it is well posed provided that the PML damping parameter and mesh size are small enough. Finally, computational results illustrating the efficiency of the finite element PML approximation are presented. © 2010 American Mathematical Society.en
dc.description.sponsorshipThis work was supported in part by the National Science Foundation throughGrant DMS-0609544 and in part by award number KUS-C1-016-04 made by KingAbdulla University of Science and Technology (KAUST).en
dc.publisherAmerican Mathematical Society (AMS)en
dc.subjectElastic wave problemen
dc.subjectElastic waves scatteringen
dc.subjectHelmholtz equationen
dc.subjectPML layeren
dc.titleAnalysis of a finite PML approximation to the three dimensional elastic wave scattering problemen
dc.typeArticleen
dc.identifier.journalMathematics of Computationen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
kaust.grant.numberKUS-C1-016-04en
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