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

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