  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, piecewise-linear, 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.
