Hardy Space Infinite Elements Research Articles

Two scale Hardy space infinite elements for scalar waveguide problems

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.

Convergence of Hardy Space Infinite Elements for Helmholtz Scattering and Resonance Problems

We perform a convergence analysis for discretizations of Helmholtz scattering and resonance problems obtained by Hardy space infinite elements. Superalgebraic convergence rates with respect to the number of Hardy space degrees of freedom are achieved. We consider spheres and piecewise polytopes as transparent boundaries. The analysis is based on a G\aa rding-type inequality and standard operator theoretical results. While the obtained results are related to those for the radial perfectly matched layer method, different techniques are used in the analysis.

Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems

We consider time-harmonic linear elasticity equations in domains containing two-dimensional semi-infinite strips. Since for such problems there exist modes with different signs of group and phase velocity, standard perfectly matched layer (PML) as well as standard Hardy space infinite element methods fail.We apply a recently developed infinite element method for a physically correct discretization of such waveguide 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 certain Hardy spaces. The Hardy space is chosen such that the construction of a simple infinite element is possible.The method does not use a modal separation and works on domains of frequencies. On those domains the involved operators are frequency independent and hence lead to linear eigenvalue problems when computing resonances. Numerical experiments containing convergence tests and resonance problems are included.

Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs

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.

Convergence of infinite element methods for scalar waveguide problems

We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the Perfectly Matched Layer (PML) method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments.

Exact Sequences of High Order Hardy Space Infinite Elements for Exterior Maxwell Problems

This paper describes the construction of a new family of prismatic infinite elements for electromagnetic scattering and resonance problems. The construction is based on the use of the pole condition to define the equivalent of an “outgoing wave.” Specifically, a function satisfies the pole condition if and only if a certain transformation of this function belongs to a Hardy space. We use tensor products of cochain complexes to obtain four different infinite element spaces which form an exact sequence corresponding to the deRham complex in the exterior domain. Numerical tests indicate superalgebraic convergence in the number of additional unknowns per degree of freedom on the coupling boundary.

Hardy space infinite elements for Helmholtz-type problems with unbounded inhomogeneities

This paper introduces a class of approximate transparent boundary conditions for the solution of Helmholtz-type resonance and scattering problems on unbounded domains. The computational domain is assumed to be a polygon. A detailed description of two variants of the Hardy space infinite element method which relies on the pole condition is given. The method can treat waveguide-type inhomogeneities in the domain with non-compact support. The results of the Hardy space infinite element method are compared to a perfectly matched layer method. Numerical experiments indicate that the approximation error of the Hardy space decays exponentially in the number of Hardy space modes.

Hardy Space Infinite Elements for Scattering and Resonance Problems

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.