eprintid: 10204451 rev_number: 9 eprint_status: archive userid: 699 dir: disk0/10/20/44/51 datestamp: 2025-02-10 10:33:24 lastmod: 2025-02-10 10:33:24 status_changed: 2025-02-10 10:33:24 type: article metadata_visibility: show sword_depositor: 699 creators_name: Caetano, António M creators_name: Chandler-Wilde, Simon N creators_name: Claeys, Xavier creators_name: Gibbs, Andrew creators_name: Hewett, David P creators_name: Moiola, Andrea title: Integral equation methods for acoustic scattering by fractals ispublished: pub divisions: UCL divisions: B04 divisions: C06 divisions: F59 keywords: Helmholtz equation, function spaces, iteratedfunction system, Galerkin method, boundaryelement method note: © 2025 The Author(s). Published by the Royal Society under the terms of theCreative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/). abstract: 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. date: 2025-01 date_type: published publisher: The Royal Society official_url: https://doi.org/10.1098/rspa.2023.0650 oa_status: green full_text_type: pub language: eng primo: open primo_central: open_green verified: verified_manual elements_id: 2358835 doi: 10.1098/rspa.2023.0650 lyricists_name: Hewett, David lyricists_id: DHEWE35 actors_name: Hewett, David actors_id: DHEWE35 actors_role: owner full_text_status: public publication: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences volume: 481 number: 2306 article_number: 20230650 citation: Caetano, António M; Chandler-Wilde, Simon N; Claeys, Xavier; Gibbs, Andrew; Hewett, David P; Moiola, Andrea; (2025) Integral equation methods for acoustic scattering by fractals. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 481 (2306) , Article 20230650. 10.1098/rspa.2023.0650 <https://doi.org/10.1098/rspa.2023.0650>. Green open access document_url: https://discovery.ucl.ac.uk/id/eprint/10204451/7/Hewett_Integral%20equation%20methods%20for%20acoustic%20scattering%20by%20fractals_VoR.pdf