The Complex-Scaled Half-Space Matching Method

TL;DR

This paper introduces a novel complex-scaled half-space matching method for 2D scattering problems, ensuring well-posedness for real wavenumbers and enabling exponentially decaying solutions for improved computational efficiency.

Contribution

It combines the HSM framework with complex-scaling to create a new, well-posed formulation for real wavenumbers, enhancing numerical stability and decay properties.

Findings

Method is provably well-posed for real wavenumbers

Solutions decay exponentially at infinity, improving computation

Preliminary numerical results validate effectiveness

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 of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.