First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker-Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763-1784) to solve the time-fractional Fokker-Planck equation on a domain Ω × [0,T] with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the L2(Ω) norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative α approaches the classical value of 1. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker-Planck problem, we present a new L2(Ω) convergence proof that avoids a flaw in the analysis of Le et al.'s paper for the semidiscrete (backward Euler scheme in time) method.
- finite elements
- fractional Fokker-Planck equation
- Gronwall inequality
- initial-boundary value problem
- time-dependent forcing