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Integral equations and boundary element methods for rough surface scattering

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MWSW03 - Computational methods for multiple scattering

Integral equation methods are popular for the computational simulation of problems of time harmonic wave scattering by (unbounded) rough surfaces, but the analysis of numerical methods faces challenges, particularly for real wavenumbers. These include but are not limited to: i) the possible existence of guided waves that are solutions of the homogeneous problem that are localised near the rough surface (these often termed trapped waves when the surface is periodic); ii) that the standard single- and double-layer integral operators are not bounded operators on unbounded surfaces (for real wavenumbers); iv) that standard Galerkin BEM analysis techniques, which rely on establishing that the operator is a compact perturbation of a coercive operator are problematic when the surface is unbounded; v) that the infinite rough surface has to be truncated to a “finite section” for practical computation. In this talk we discuss probably the simplest problem of this class, acoustic scattering by a sound soft rough surface that is the graph of a bounded, Lipschitz continuous function. Inspired by Spence et al. (Comm. Pure Appl. Math. 2011) and Chandler-Wilde & Spence (arXiv:2210.02432 2022) we show that it is possible to write down a continuous and coercive second kind integral equation formulation for this problem, that reduces to the combined field integral equation of Chandler-Wilde, Heinemeyer, and Potthast (Proc. R. Soc. A 2006 ) when the boundary is flat. This coercivity ensures stablity of every Galerkin BEM and of the finite-section method, and leads to a bound for the convergence of GMRES , that the number of GMRES iterations needed to obtain a given accuracy is bounded independently of the mesh size and of the size of the finite section. We illustrate the theory with 2D numerical results. This is joint work with Martin Averseng and Euan Spence, University of Bath. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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