Integral equation methods for acoustic scattering by fractals

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• Dave Hewett (UCL)
• Tuesday 14 November 2023, 14:00-15:00
• Nuffield G13.

If you have a question about this talk, please contact Sergey Sergeev.

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer Γ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on Γ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when Γ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When Γ is uniformly of d-dimensional Hausdorff dimension in a sense we make precise (a d-set), the operator in our equation is an integral operator on Γ with respect to d-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When Γ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on Γ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code.

This is joint work with António Caetano (Aveiro), Simon Chandler-Wilde (Reading), Xavier Claeys (Sorbonne), Andrew Gibbs (UCL) and Andrea Moiola (Pavia).

This talk is part of the Optimisation and Numerical Analysis Seminars series.

This talk is included in these lists:

• Nuffield G13
• Optimisation and Numerical Analysis Seminars
• School of Mathematics events

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