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

Kim Ngan Le, Martin Stynes

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Abstract

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 Rd 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 uh(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).

Original languageEnglish
Article number22
Number of pages16
JournalJournal of Scientific Computing
Volume86
Issue number2
DOIs
Publication statusPublished - Feb 2021

Keywords

  • Fokker–Planck equation
  • Variable-order fractional derivative
  • α-robust

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