  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.
