A Hausdorff measure boundary element method for acoustic scattering by fractal screens
We introduce and analyse a novel discretization method for acoustic scattering by fractal screens. In contrast to previous studies, in which a conventional boundary element method (BEM) was applied on a “pre-fractal” approximation of the fractal, here we work with BEM basis functions supported on the fractal itself, integrating with respect to Hausdorff measure rather than the conventional surface (Lebesgue) measure. Using some old and new function space results we prove convergence of our BEM, and obtain convergence rates under natural solution regularity assumptions. We also detail a strategy for the numerical evaluation of the required Hausdorff measure integrals, accompanied by a fully discrete convergence analysis. This is joint work with Simon Chandler-Wilde (Reading), António Caetano (Averio), Andrew Gibbs (UCL) and Andrea Moiola (Pavia).
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