Variational Formulations for the Determination of Resonant States in Scattering Problems

Consider the scattering of an acoustic wave by a rigid obstacle. The poles of the analytical continuation of the resolvent operator are called scattering frequencies. On their localization depend the time-decay of the solution and the location of the energy peaks of the steady-state solution. Two methods are proposed to construct explicitly the analytical continuation of the resolvent: the localized finite element method or the coupling between variational formulation and integral representation, which both rely upon the reduction of the exterior Helmholtz problem to a bounded domain. The determination of the scattering frequencies then amounts to solving a nonlinear eigenvalue problem for a completely continuous operator. Then, the expansion of the approximate steady-state solution in the vicinity of a scattering frequency is computed. Numerical results for a simple one-dimensional problem are presented.