An α-Robust Semidiscrete Finite Element Method for a Fokker–Planck Initial-Boundary Value Problem with Variable-Order Fractional Time Derivative - Le, Kim-Ngan; Stynes, Martin (Journal of scientific computing, 2021)
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 - u h ) ( t ) ‖ L 2 ( Ω ) and ∫ 0 t ∂ t 1 - α ( u - u h ) ( s ) 1 2 d s are proved for each t ∈ [ 0 , T ] under the assumptions that the following quantities are finite: ‖ u ( · , 0 ) ‖ H 2 ( Ω ) , | u ( · , t ) | H 1 ( Ω ) for each t , and ∫ 0 t [ ‖ u ( · , t ) ‖ H 2 ( Ω ) 2 + | ∂ t 1 - α u | H 2 ( 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).