Numerical Quadrature for Singular Integrals on Fractals

12/22/2021

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by   A. Gibbs, et al.

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We present and analyse numerical quadrature rules for evaluating smooth and singular integrals on self-similar fractal sets. The integration domain Γ⊂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 the Hausdorff measure ℋ^d, where d is the Hausdorff dimension of Γ, and both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Γ are decomposed into sums of integrals over suitable partitions of Γ into self-similar sub-components. For certain singular integrands of logarithmic or algebraic type we show how in the context of such a partitioning the self-similarity of Γ 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 sub-components. 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.