We propose and analyze a mixed finite element method for the spatial approximation of a time-fractional Fokker--Planck equation in a convex polyhedral domain, where the given driving force is a function of space. Taking into account the limited smoothing properties of the model, and considering an appropriate splitting of the errors, we employed a sequence of clever energy arguments to show optimal convergence rates with respect to both approximation properties and regularity results. In particular, error bounds for both primary and secondary variables are derived in $L^2$-norm for cases with smooth and nonsmooth initial data. We further investigate a fully implicit time-stepping scheme based on a convolution quadrature in time generated by the backward Euler method. Our main result provides pointwise-in-time optimal $L^2$-error estimates for the primary variable. Numerical examples are then presented to illustrate the theoretical contributions.

翻译：本文针对凸多面体域上一类时间分数阶Fokker-Planck方程的空间逼近，提出并分析了一种混合有限元方法，其中给定的驱动力为空间函数。考虑到该模型有限的平滑性质，并通过对误差进行适当分解，我们运用一系列巧妙的能量论证，在逼近性质和正则性结果两个层面证明了最优收敛速率。具体而言，针对光滑与非光滑初值情形，本文推导了主变量与次变量在$L^2$范数下的误差界。进一步地，我们研究了基于后向欧拉法生成的卷积时间求积的全隐式时间步进格式。主要结果给出了主变量在时间上的逐点最优$L^2$误差估计。最后通过数值算例验证了理论结论。