The Complex-Scaled Half-Space Matching Method
Anne-Sophie Bonnet-Ben Dhia, Simon N. Chandler‐Wilde, Sonia Fliss, Christophe Hazard, Karl‐Mikael Perfekt, Yohanes Tjandrawidjaja
Abstract
The Half-Space Matching (HSM) method has recently been developed as a new\nmethod for the solution of 2D scattering problems with complex backgrounds,\nproviding an alternative to Perfectly Matched Layers (PML) or other artificial\nboundary conditions. Based on half-plane representations for the solution, the\nscattering problem is rewritten as a system of integral equations in which the\nunknowns are restrictions of the solution to the boundaries of a finite number\nof overlapping half-planes contained in the domain: this integral equation\nsystem is coupled to a standard finite element discretisation localised around\nthe scatterer. While satisfactory numerical results have been obtained for real\nwavenumbers, wellposedness and equivalence to the original scattering problem\nhave been established only for complex wavenumbers. In the present paper, by\ncombining the HSM framework with a complex-scaling technique, we provide a new\nformulation for real wavenumbers which is provably well-posed and has the\nattraction for computation that the complex-scaled solutions of the integral\nequation system decay exponentially at infinity. The analysis requires the\nstudy of double-layer potential integral operators on intersecting infinite\nlines, and their analytic continuations. The effectiveness of the method is\nvalidated by preliminary numerical results.\n