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, 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. Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

翻译：本文考虑一类在自相似分形集上相对于一对自相似测度的二重积分的数值计算，其中被积函数具有对数型或代数型的弱奇异性。最近的一项研究[Gibbs, Hewett and Moiola, Numer. Alg., 2023]表明，当分形集在某种意义上是“不相交”的（例如康托尔集）时，利用测度的自相似性和被积函数的齐次性质，可以将奇异积分精确地表示为正则积分，从而便于进行数值逼近。本文提出了一种方法，将上述结果推广到分形集非不相交的情形。我们的方法适用于许多经典例子，包括谢尔宾斯基三角形、维切克分形、谢尔宾斯基地毯和科赫雪花。