Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem
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- by James H. Bramble, Joseph E. Pasciak and Dimitar Trenev PDF
- Math. Comp. 79 (2010), 2079-2101 Request permission
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
- Email: bramble@math.tamu.edu
- Joseph E. Pasciak
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: pasciak@math.tamu.edu
- Dimitar Trenev
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: trenev@math.tamu.edu
- Received by editor(s): October 30, 2008
- Received by editor(s) in revised form: July 15, 2009
- Published electronically: April 19, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 2079-2101
- MSC (2010): Primary 65F10, 78M10, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-10-02355-0
- MathSciNet review: 2684356