On the half-space matching method for real wavenumber

TL;DR

This paper proves the equivalence and well-posedness of the Half-Space Matching method for 2D scattering problems with real wavenumbers, extending previous results from complex wavenumbers and ensuring its applicability.

Contribution

The paper demonstrates that the HSM formulation is equivalent to the original scattering problem for real wavenumbers under radiation conditions, establishing its well-posedness.

Findings

HSM is equivalent to the original problem for real wavenumbers.

The radiation condition ensures well-posedness of the HSM formulation.

A new radiation condition relates boundary traces to Sommerfeld conditions.

Abstract

The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretisation localised around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem have been established only for complex wavenumbers. In the present paper we show, in the case of a homogeneous background, that the HSM formulation is equivalent to the original scattering problem also for real wavenumbers, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that, if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering.