TY  - JOUR
UR  - https://doi.org/10.1098/rspa.2023.0650
PB  - The Royal Society
ID  - discovery10204451
N2  - We study sound-soft time-harmonic acousticscattering by general scatterers, including fractalscatterers, in 2D and 3D space. For an arbitrarycompact scatterer ? we reformulate the Dirichletboundary value problem for the Helmholtz equationas a first kind integral equation (IE) on ? involvingthe Newton potential. The IE is well-posed, exceptpossibly at a countable set of frequencies, andreduces to existing single-layer boundary IEs when? is the boundary of a bounded Lipschitz open set,a screen, or a multi-screen. When ? is uniformlyof d-dimensional Hausdorff dimension in a sensewe make precise (a d-set), the operator in ourequation is an integral operator on ? 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 ? is the fractal attractor of an iteratedfunction system of contracting similarities we proveconvergence rates under assumptions on ? 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.
KW  - Helmholtz equation
KW  -  function spaces
KW  -  iteratedfunction system
KW  -  Galerkin method
KW  -  boundaryelement method
A1  - Caetano, António M
A1  - Chandler-Wilde, Simon N
A1  - Claeys, Xavier
A1  - Gibbs, Andrew
A1  - Hewett, David P
A1  - Moiola, Andrea
JF  - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
AV  - public
VL  - 481
Y1  - 2025/01//
TI  - Integral equation methods for acoustic scattering by fractals
N1  - © 2025 The Author(s). Published by the Royal Society under the terms of theCreative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/).
IS  - 2306
ER  -