A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

TL;DR

This paper introduces a novel boundary element method using Hausdorff measure for acoustic scattering by fractal screens, proving convergence and providing numerical validation for fractal geometries with complex dimensions.

Contribution

It is the first application of BEM with basis functions supported on fractals and integration with respect to Hausdorff measure, extending boundary element methods to fractal scatterers.

Findings

Proved convergence of the Hausdorff BEM for fractal screens with Hausdorff dimension d.

Established superconvergence for smooth functionals under regularity assumptions.

Provided numerical experiments validating theoretical convergence and regularity results.

Abstract

Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this ``Hausdorff BEM'' for acoustic scattering in $\mathbb{R}^{n+1}$ ($n=1,2$) when the scatterer, assumed to be a compact subset of $\mathbb{R}^n\times\{0\}$, is a $d$-set for some $d\in (n-1,n]$, so that, in particular, the scatterer has Hausdorff dimension $d$. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in $\mathbb{R}^2$ by Cantor sets, and in $\mathbb{R}^3$ by Cantor dusts.