  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.
