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

Abstract

A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain $\Omega \times [0,T]$, where $\Omega$ is an open bounded domain in $\mathbb {R}^d$ for some $d\ge 1$, and the order $\alpha (x)$ of the Riemann-Liouville fractional derivative may vary in space with $1/2< \alpha (x) < 1$ 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 $\Vert (u - u_h)(t)\Vert _{L^2(\Omega )}$ and $\int _0^t \left| \partial _t^{1-\alpha } (u-u_h)(s)\right| _1^2 \,ds$ are proved for each $t\in [0,T]$ under the assumptions that the following quantities are finite: $\Vert u(\cdot , 0)\Vert _{H^2(\Omega )}, |u(\cdot , t)|_{H^1(\Omega )}$ for each t, and $\int _0^t [\Vert u(\cdot , t)\Vert _{H^2(\Omega )}^2 + |\partial _t^{1-\alpha }u|_{H^2(\Omega )}^2]$, where u(x, t) is the unknown solution. Furthermore, these error estimates are $\alpha$-robust: they do not fail when $\alpha \rightarrow 1$, 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).