In this work, based on the complete Bernstein function, we propose a generalized regularity analysis including maximal $\mathrm{L}^p$ regularity for the Fokker--Planck equation, which governs the subordinated Brownian motion with the inverse tempered stable subordinator that has a drift. We derive a generalized time--stepping finite element scheme based on the backward Euler convolution quadrature, and the optimal-order convergence of the numerical solutions is established using the proven solution regularity. Further, the analysis is generalized to more general diffusion equations. Numerical experiments are provided to support the theoretical results.

翻译：在这项工作中,基于完整的伯恩斯坦函数,我们建议进行一项普遍常规分析,包括Fokker-Planck等式的最高值$/mathrm{L ⁇ p$常规值,该等式以反向温和稳定的副协调员管理从属的布朗恩运动,该等式具有漂移性。我们根据后向的电动振动四分法制定了一个普遍的时间分步的有限元素计划,并且利用经证明的解决方案的常规性,确定了数字解决方案的最佳顺序趋同。此外,该分析被普遍推广到更普遍的传播方程。提供了数字实验来支持理论结果。