Hardy Space Infinite Elements for Scattering and Resonance Problems

Abstract

This paper introduces a new type of infinite element for scattering and resonance problems that is derived from a variant of the pole condition as radiation condition. This condition states that a certain transform of the exterior solution belongs to the Hardy space of $L^2$ boundary values of holomorphic functions on the unit disc if and only if the solution is outgoing. We obtain a symmetric variational formulation of the problem in this Hardy space. Our infinite elements correspond to a Galerkin discretization with respect to the standard monomial orthogonal basis of this Hardy space and lead to simple element matrices. Hardy space infinite elements are particularly well suited for solving resonance problems since they preserve the eigenvalue structure of the problem. We prove superalgebraic convergence for a separated problem. Numerical experiments exhibit fast convergence over a wide range of wave numbers.