Abstract:
In this talk, 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 trans-
formed 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.
This talk is a joint work with Seungil Kim.
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.