A New Gradient Scheme of a Time Fractional Fokker–Planck Equation with Time Independent Forcing and Its Convergence Analysis

Abstract

We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the time fractional Fokker–Planck equation with time independent forcing in any space dimension. Using [5] which dealt with GDM for linear advection problems, we develop a new fully discrete implicit GS (Gradient Scheme) for the stated model. We prove new discrete a priori estimates which yield estimates on the discrete solution in $L^\infty (L^2)$ and $L^2(H^1)$ discrete norms. Thanks to these discrete a priori estimates, we prove new error estimates in the discrete norms of $L^\infty (L^2)$ and $L^2(H^1)$. The main ingredients in the proof of these error estimates are the use of the stated discrete a priori estimates and a comparison with some well chosen auxiliary schemes. These auxiliary schemes are approximations of convective-diffusive elliptic problems in each time level. We state without proof the convergence analysis of these auxiliary schemes. Such proof uses some adaptations of the [6, Proof of Theorem 2.28] dealt with GDM for the case of elliptic diffusion problems. These results hold for all the schemes within the framework of GDM. This work can be viewed as an extension to our recent one [2].