Acoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximation

TL;DR

This paper analyzes acoustic scattering by impedance screens with fractal boundaries, establishing well-posedness, boundary integral formulations, and convergence of boundary element methods, supported by numerical validation for fractal shapes.

Contribution

It introduces a framework for well-posedness and boundary element approximation of scattering problems involving fractal boundaries, extending previous smooth-boundary results.

Findings

Boundary integral operators are coercive with compact perturbations.

Boundary element methods converge on prefractal screens to fractal solutions.

Numerical results validate theoretical convergence for Koch snowflake shapes.

Abstract

We study time-harmonic scattering in $\mathbb{R}^n$ ($n=2,3$) by a planar screen (a "crack" in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset $\Gamma$ of the hyperplane $\mathbb{R}^{n-1}\times \{0\}$, on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, $\Gamma$ can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such $\Gamma$ we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when $\partial\Gamma$ is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother "prefractal" screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake.