  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.
