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).
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K.-N. Le: The research of this author is supported in part by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (Project Number DP170100605).
M. Stynes: The research of this author is supported in part by the National Natural Science Foundation of China under Grant NSAF-U1930402.
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Le, KN., Stynes, M. An \(\alpha \)-Robust Semidiscrete Finite Element Method for a Fokker–Planck Initial-Boundary Value Problem with Variable-Order Fractional Time Derivative. J Sci Comput 86, 22 (2021). https://doi.org/10.1007/s10915-020-01375-x
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DOI: https://doi.org/10.1007/s10915-020-01375-x