Boundary element methods for acoustic scattering by fractal screens

TL;DR

This paper develops boundary element methods to simulate acoustic scattering by fractal screens, proving convergence of numerical solutions and demonstrating their effectiveness through numerical examples.

Contribution

It introduces a novel approach for approximating fractal screens with prefractals and establishes convergence conditions for boundary element methods in this context.

Findings

Proved convergence of boundary element solutions to fractal screen scattering problems.

Established mesh size conditions ensuring accurate approximations.

Validated theoretical results with numerical simulations.

Abstract

We study boundary element methods for time-harmonic scattering in $\mathbb{R}^n$ ($n=2,3$) by a fractal planar screen, assumed to be a non-empty bounded subset $\Gamma$ of the hyperplane $\Gamma_\infty=\mathbb{R}^{n-1}\times \{0\}$. We consider two distinct cases: (i) $\Gamma$ is a relatively open subset of $\Gamma_\infty$ with fractal boundary (e.g.\ the interior of the Koch snowflake in the case $n=3$); (ii) $\Gamma$ is a compact fractal subset of $\Gamma_\infty$ with empty interior (e.g.\ the Sierpinski triangle in the case $n=3$). In both cases our numerical simulation strategy involves approximating the fractal screen $\Gamma$ by a sequence of smoother "prefractal" screens, for which we compute the scattered field using boundary element methods that discretise the associated first kind boundary integral equations. We prove sufficient conditions on the mesh sizes guaranteeing convergence to the limiting fractal solution, using the framework of Mosco convergence. We also provide numerical examples illustrating our theoretical results.