We consider computation of resonances in open systems and acoustic scattering
problems. These problems are posed on an unbounded domain and domain truncation
is required for the numerical computation. In this paper, a perfectly matched layer
(PML) technique is proposed for computation of solutions to the unbounded domain
problems.
For resonance problems, resonance functions are characterized as improper eigenfunction
(non-zero solutions of the eigenvalue problem which are not square integrable)
of the Helmholtz equation on an unbounded domain. We shall see that the
application of the spherical PML converts the resonance problem to a standard eigenvalue
problem on the infinite domain. Then, the goal will be to approximate the eigenvalues
first by replacing the infinite domain by a finite computational domain with
a convenient boundary condition and second by applying finite elements to the truncated
problem. As approximation of eigenvalues of problems on a bounded domain
is classical [12], we will focus on the convergence of eigenvalues of the (continuous)
PML truncated problem to those of the infinite PML problem. Also, it will be shown
that the domain truncation does not produce spurious eigenvalues provided that the
size of computational domain is sufficiently large.
The spherical PML technique has been successfully applied for approximation
of scattered waves [13]. We develop an analysis for the case of a Cartesian PML application to the acoustic scattering problem, i.e., solvability of infinite and truncated
Cartesian PML scattering problems and convergence of the truncated Cartesian PML
problem to the solution of the original solution in the physical region as the size of
computational domain increases.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2009-08-2931 |
| Date | 14 January 2010 |
| Creators | Kim, Seungil |
| Contributors | Pasciak, Joseph E. |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation |
| Format | application/pdf |
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