We present and analyse numerical quadrature rules for evaluating smooth and singular integrals on self-similar fractal sets. The integration domain $\Gamma\subset\mathbb{R}^n$ 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 the Hausdorff measure $\mathcal{H}^d$, where $d$ is the Hausdorff dimension of $\Gamma$, and 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 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 $\Gamma$ 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.
翻译:我们提出并分析对自相似分形集成进行平滑和单元集成的评估的数值二次方块规则。 集成域 $\Gamma\ subset\ mathb{R ⁇ n$ 假设是连接符合开放设置条件的相似点的迭代功能系统的紧凑吸引者。 集成涉及Hausdorf 测量 $\mathcal{H ⁇ d$, 美元是 $\Gamma$的Husdorf 维度, 考虑单元和双元集成。 我们的重点是复合方块规则, 将$\Gamma$的集成分成分解成一个组合体块的组合体块块块块, 以美元和双倍集成成成成成份法的组合体块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块图,, 对于正块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块块