%0 Journal Article
%A Caetano, António M
%A Chandler-Wilde, Simon N
%A Claeys, Xavier
%A Gibbs, Andrew
%A Hewett, David P
%A Moiola, Andrea
%D 2025
%F discovery:10204451
%I The Royal Society
%J Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
%K Helmholtz equation, function spaces, iteratedfunction system, Galerkin method, boundaryelement method
%N 2306
%T Integral equation methods for acoustic scattering by fractals
%U https://discovery.ucl.ac.uk/id/eprint/10204451/
%V 481
%X We study sound-soft time-harmonic acousticscattering by general scatterers, including fractalscatterers, in 2D and 3D space. For an arbitrarycompact scatterer Γ we reformulate the Dirichletboundary value problem for the Helmholtz equationas a first kind integral equation (IE) on Γ involvingthe Newton potential. The IE is well-posed, exceptpossibly at a countable set of frequencies, andreduces to existing single-layer boundary IEs whenΓ is the boundary of a bounded Lipschitz open set,a screen, or a multi-screen. When Γ is uniformlyof d-dimensional Hausdorff dimension in a sensewe make precise (a d-set), the operator in ourequation is an integral operator on Γ with respectto d-dimensional Hausdorff measure, with kernel theHelmholtz fundamental solution, and we proposea piecewise-constant Galerkin discretization of theIE, which converges in the limit of vanishing meshwidth. When Γ is the fractal attractor of an iteratedfunction system of contracting similarities we proveconvergence rates under assumptions on Γ and the IEsolution, and describe a fully discrete implementationusing recently proposed quadrature rules for singularintegrals on fractals. We present numerical results fora range of examples and make our software availableas a Julia code.
%Z © 2025 The Author(s). Published by the Royal Society under the terms of theCreative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/).