The computation of resonances in open systems
using a perfectly matched layer
by S. Kim and J.E. Pasciak.
ABSTRACT:
In this paper, we consider the problem of computing resonances in open systems.
We first characterize resonances in terms of (improper) eigenfunctions
of the Helmholtz
operator on an unbounded domain. The perfectly matched layer (PML)
technique has been
successfully applied to the computation of scattering problems.
We shall see that
the application of PML converts the resonance problem to a standard eigenvalue
problem (still on an infinite domain). This new eigenvalue problem involves an
operator which resembles the original Helmholtz equation transformed by
a complex
shift in coordinate system. Our goal will be to approximate
the shifted operator
first by replacing the infinite domain by a finite (computational)
domain
with a convenient boundary
condition and second by applying finite
elements on the computational domain. We shall
prove that the first of these steps leads to eigenvalue
convergence (to the desired resonance values)
which is free from spurious computational eigenvalues provided
that the size of computational
domain is sufficiently large. The analysis of the second step is
classical.
Finally, we illustrate the behavior
of the method applied to numerical experiments in one and
two spatial dimensions.
ACKNOWLEGEMENTS:
This work was supported in part by the National Science
Foundation
through grant number DMS-0609544.