The Complex-Scaled Half-Space Matching Method
Pages 512 - 557
Abstract
The half-space matching (HSM) method has recently been developed as a new method for the solution of two-dimensional scattering problems with complex backgrounds, providing an alternative to perfectly matched layers 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 discretization localized around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness 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.
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© 2022, Society for Industrial and Applied Mathematics.
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Published: 01 January 2022
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