Numerical Quadrature for Singular Integrals on Fractals

TL;DR

This paper develops and analyzes numerical quadrature methods for evaluating both regular and singular integrals on self-similar fractal sets, enabling accurate computations in fractal boundary element methods for acoustic scattering.

Contribution

It introduces composite quadrature rules leveraging measure invariance to exactly evaluate certain singular integrals on fractals, with proven second-order convergence for regular integrals.

Findings

Exact expression of singular integrals via regular integrals using invariance.

Second-order convergence of the barycentre rule for regular integrals.

Application to accurate evaluation of singular integrals in fractal boundary element methods.

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.