We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper [Gibbs, Hewett and Moiola, Numer. Alg., 2023] it was shown that when the fractal set is "disjoint" in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

翻译：我们考虑一类在自相似分形集（迭代函数系统的吸引子）上关于一对自相似测度的二重积分的数值求值，其中被积函数为对数型或代数型的弱奇异函数。在近期论文[Gibbs, Hewett and Moiola, Numer. Alg., 2023]中已证明，当该分形集在特定意义下“不相交”（例如康托尔集）时，可利用测度的自相似性结合被积函数的齐次性，将奇异积分精确表示为易于数值逼近的正则积分。本文提出一种方法，将这些结果推广至分形集非不相交但非重叠（即满足开集条件）的情形。该方法适用于众多经典实例，包括谢尔宾斯基三角形、维切克分形、谢尔宾斯基地毯及科赫雪花。