Integral equation methods for acoustic scattering by fractals

• Main Author: Caetano, António M.
• Publication Date: 2025
• Other Authors: Chandler-Wilde, Simon N., Claeys, Xavier, Gibbs, Andrew, Hewett, David P., Moiola, Andrea
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10773/44923
• topic: Helmholtz equation; Function spaces; Iterated function system; Galerkin method; Boundary element method

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