Abstract

In part 1 (the prequel to this talk), it will be shown that sound-soft scattering by fractal screens (such as the Cantor Set or Sierpinski Triangle) can be modelled by generalising the Boundary Element Method to obstacles with non-integer dimension, the so-called “Hausdorff BEM ”. In Hausdorff BEM , like the scatterer, the mesh elements are self-similar fractals with non-integer dimension and zero Lebesgue measure. Therefore, implementation of this non-standard Galerkin BEM presents an interesting new challenge: approximation of double integrals of singular Green’s kernels, over fractal domains with respect to Hausdorff measure. This motivated the research I will present in this talk. I will begin by discussing numerical methods for approximating smooth integrals with respect to Hausdorff measure, summarising our recent contributions and existing methods. Next, for integrals over a self-similar domain, where the integrand has a singularity of logarithmic or algebraic type, I will present a novel algorithm which exploits this self-similarity to reformulate this singular integral as a sum of smooth integrals (which are easier to approximate). This technique forms an essential part of our Hausdorff BEM implementation. I will conclude by presenting fully discrete estimates for Hausdorff BEM , alongside numerical results.