Numerical Quadrature for Singular Integrals on Fractals

Abstract

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $\mathbb{R}^n$ is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any ``invariant'' (also known as ``balanced'' or ``self-similar'') measure supported on $\Gamma$, including in particular the Hausdorff measure $\mathcal{H}^d$ restricted to $\Gamma$, where $d$ is the Hausdorff dimension of $\Gamma$. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over $\Gamma$ are decomposed into sums of integrals over suitable partitions of $\Gamma$ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens.