An α -Robust Semidiscrete Finite Element Method for a Fokker–Planck Initial-Boundary Value Problem with Variable-Order Fractional Time Derivative

A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain Ω × [0 , T] , where Ω is an open bounded domain in R^d for some d≥ 1 , and the order α(x) of the Riemann-Liouville fractional derivative may vary in space with 1 / 2 < α(x) < 1 for all x. Such problems appear naturally in the formulation of certain continuous-time random walk models. Uniqueness of any solution u of the problem is proved under reasonable hypotheses. A semidiscrete numerical method, using finite elements in space to yield a solution u_h(t) , is constructed. Error estimates for ‖(u-uh)(t)‖L2(Ω) and ∫0t|∂t1-α(u-uh)(s)|12ds are proved for each t∈ [0 , T] under the assumptions that the following quantities are finite: ‖u(·,0)‖H2(Ω),|u(·,t)|H1(Ω) for each t, and ∫0t[‖u(·,t)‖H2(Ω)2+|∂t1-αu|H2(Ω)2], where u(x, t) is the unknown solution. Furthermore, these error estimates are α-robust: they do not fail when α→ 1 , the classical Fokker–Planck problem. Sharper results are obtained for the special case where the drift term of the problem is not present (which is of interest in certain applications).