Integral equation methods for acoustic scattering by fractals

• DOI: 10.1098/rspa.2023.0650
• Publication Date: 2025-01-30T15:05:04Z
• Authors: António M Caetano; Simon N Chandler-...; Xavier Claeys; Andrew Gibbs; David P Hewett; Andrea Moiola
• Keywords: 28A80, 65R20, 78A45; FOS: Mathematics; Mathematics - Numerical Analysis; Numerical Analysis (math.NA)

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal in 2D and 3D space. For an arbitrary compact scatterer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mi mathvariant="italic">Γ</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> we reformulate the Dirichlet boundary value problem for Helmholtz equation as a first kind integral (IE) on involving Newton potential. The IE is well-posed, except possibly at countable set of frequencies, reduces to existing single-layer IEs when bounded Lipschitz open set, screen, or multi-screen. When uniformly <mml:mi>d</mml:mi> -dimensional Hausdorff dimension sense make precise (a -set), operator our with respect measure, kernel fundamental solution, propose piecewise-constant Galerkin discretization IE, which converges limit vanishing mesh width. attractor iterated function system contracting similarities prove convergence rates under assumptions describe fully discrete implementation using recently proposed quadrature rules singular integrals fractals. present numerical results range examples software available Julia code.