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.

翻译：声软分形屏即使具有零表面测度，仍能散射声波。为解决此类散射问题，我们首次将边界元方法（BEM）应用于分形集支撑的基函数，其中BEM矩阵形成的积分涉及非整数阶Hausdorff测度，而非通常的（Lebesgue）表面测度。利用分形上函数空间的最新结果，我们证明了当散射体（假定为$\mathbb{R}^n\times\{0\}$的紧子集）是某个$d\in (n-1,n]$的$d$-集（即散射体具有Hausdorff维数$d$）时，该"Hausdorff BEM"的Galerkin格式在$\mathbb{R}^{n+1}$（$n=1,2$）声散射问题中的收敛性。对于迭代函数系统吸引子的一类分形，在底层边界积分方程解的特定正则性假设下，我们证明了Hausdorff BEM的收敛速率及光滑反线性泛函的超收敛性。进一步，我们提出了实现Hausdorff BEM的数值求积程序，并通过数值（Hausdorff测度）积分估计和分形逆估计（用于估计离散条件数）完成了全离散收敛性分析。最后，数值实验验证了理论结果的精确性及解正则性假设，包括Cantor集在$\mathbb{R}^2$中的散射及Cantor尘在$\mathbb{R}^3$中的散射结果。