@article{discovery10204451,
           month = {January},
            year = {2025},
          number = {2306},
           title = {Integral equation methods for acoustic scattering by fractals},
          volume = {481},
            note = {{\copyright} 2025 The Author(s). Published by the Royal Society under the terms of theCreative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/).},
       publisher = {The Royal Society},
         journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
        abstract = {We study sound-soft time-harmonic acousticscattering by general scatterers, including fractalscatterers, in 2D and 3D space. For an arbitrarycompact scatterer {\ensuremath{\Gamma}} we reformulate the Dirichletboundary value problem for the Helmholtz equationas a first kind integral equation (IE) on {\ensuremath{\Gamma}} involvingthe Newton potential. The IE is well-posed, exceptpossibly at a countable set of frequencies, andreduces to existing single-layer boundary IEs when{\ensuremath{\Gamma}} is the boundary of a bounded Lipschitz open set,a screen, or a multi-screen. When {\ensuremath{\Gamma}} is uniformlyof d-dimensional Hausdorff dimension in a sensewe make precise (a d-set), the operator in ourequation is an integral operator on {\ensuremath{\Gamma}} 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 {\ensuremath{\Gamma}} is the fractal attractor of an iteratedfunction system of contracting similarities we proveconvergence rates under assumptions on {\ensuremath{\Gamma}} 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.},
        keywords = {Helmholtz equation, function spaces, iteratedfunction system, Galerkin method, boundaryelement method},
             url = {https://doi.org/10.1098/rspa.2023.0650},
          author = {Caetano, Ant{\'o}nio M and Chandler-Wilde, Simon N and Claeys, Xavier and Gibbs, Andrew and Hewett, David P and Moiola, Andrea}
}