Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we first derive the fractional Fokker-Planck equation with two-scale diffusion from the L\'evy process framework, and then the fully discrete scheme is built by using the $L_{1}$ scheme for time discretization and finite element method for space. With the help of the sharp regularity estimate of the solution, we optimally get the spatial and temporal error estimates. Finally, we validate the effectiveness of the provided algorithm by extensive numerical experiments.

翻译：Fractional Fokker-Planck 等式在描述异常动态方面起着重要作用。 据我们所知,现有讨论主要侧重于涉及一个扩散操作员的这种等式。在本文中,我们首先从L\'evy进程框架以两个尺度的分布法得出分数方程式,然后完全分离的方程式则通过使用$L ⁇ 1美元计划来建立,用于时间分解和空间的有限元素方法。根据对解决方案的精确定期估计,我们最优化地获得空间和时间误差估计。最后,我们通过广泛的数字实验来验证所提供的算法的有效性。