双尺度扩散分数阶Fokker-Planck方程的数值逼近
Numerical approximations for the fractional Fokker-Planck equation with two-scale diffusion
摘要：
分数阶Fokker-Planck方程是描述反常动力学的重要方程。据我们所知，现有的讨论主要集中在这类方程涉及一个扩散算子。本文首先在Lévy过程框架下推导了具有双尺度扩散的分数阶Fokker-Planck方程，然后采用$L_{1}$格式进行时间离散，有限元法进行空间离散，建立了全离散格式.借助于解的尖锐正则性估计，我们最优地得到了空间和时间误差估计。最后，我们通过大量的数值实验验证了所提供的算法的有效性。
英文摘要：
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 Lévy 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.
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