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

Authors: James H. Bramble, Joseph E. Pasciak and Dimitar Trenev
Journal: Math. Comp. 79 (2010), 2079-2101
MSC (2010): Primary 65F10, 78M10, 65N30
Published electronically: April 19, 2010
MathSciNet review: 2684356
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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.

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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Dimitar Trenev
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Elastic wave problem, Helmholtz equation, elastic waves scattering, PML layer
Received by editor(s): October 30, 2008
Received by editor(s) in revised form: July 15, 2009
Published electronically: April 19, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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