We construct an interpolatory high-order cubature rule to compute integrals of smooth functions over self-affine sets with respect to an invariant measure. The main difficulty is the computation of the cubature weights, which we characterize algebraically, by exploiting a self-similarity property of the integral. We propose an $h$-version and a $p$-version of the cubature, present an error analysis and conduct numerical experiments.

翻译：我们构建了一种插值型高阶立方体积分法则，用于计算光滑函数在自仿射集上关于不变测度的积分。主要难点在于立方体积分权重的计算，我们通过利用积分的自相似性质，从代数角度对其进行了刻画。我们提出了立方体积分的$h$版本和$p$版本，给出了误差分析并进行了数值实验。