A semidiscrete finite element approximation of a time-fractional Fokker Planck equation with nonsmooth initial data

We present a new stability and convergence analysis for the spatial discretization of a time-fractional Fokker-Planck equation in a convex polyhedral domain, using continuous, piecewiselinear, finite elements. The forcing may depend on time as well as on the spatial variables, and the initial data may have low regularity. Our analysis uses a novel sequence of energy arguments in combination with a generalized Gronwall inequality. Although this theory covers only the spatial discretization, we present numerical experiments with a fully discrete scheme employing a very small time step, and observe results consistent with the predicted convergence behavior.