Integral equation methods for acoustic scattering by fractals
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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.
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