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We consider the numerical solution of the Helmholtz equation in domains with one infinite cylindrical waveguide. Such problems exhibit wavenumbers on different scales in the vicinity of cut-off frequencies. This leads to performance issues for non-modal methods like the perfectly matched layer or the Hardy space infinite element method. To improve the latter, we propose a two scale Hardy space infinite element method which can be optimized for wavenumbers on two different scales. It is a tensor product Galerkin method and fits into existing analysis. Up to arbitrary small thresholds it converges exponentially with respect to the number of longitudinal unknowns in the waveguide. Numerical experiments support the theoretical error bounds. [ABSTRACT FROM AUTHOR]

We consider time harmonic wave equations in cylindrical wave-guides with physical solutions for which the signs of group and phase velocities differ. The perfectly matched layer methods select modes with positive phase velocity, and hence they yield stable, but unphysical solutions for such problems. We derive an infinite element method for a physically correct discretization of such wave-guide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are Hardy spaces of a curved domain. The Hardy space is constructed such that it contains a simple and convenient Riesz basis with small condition numbers. In this paper the new method is only discussed for a one-dimensional fourth order model problem. Exponential convergence is shown. The method does not use a modal separation and works on an interval of frequencies. Numerical experiments confirm exponential convergence. [ABSTRACT FROM AUTHOR]

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 L² 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. [ABSTRACT FROM AUTHOR]