Integral equation methods for acoustic scattering by fractals

Caetano, A. M., Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283, Claeys, X., Gibbs, A., Hewett, D. P. and Moiola, A. (2025) Integral equation methods for acoustic scattering by fractals. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 481 (2306). ISSN 1471-2946 doi: 10.1098/rspa.2023.0650

Abstract/Summary

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $\Gamma$ 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 $\Gamma$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $\Gamma$ 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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/117651
Identification Number/DOI	10.1098/rspa.2023.0650
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Royal Society Publishing
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