Abstract
An initial-boundary value problem of the form \({D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f\) is considered, where \({D}_{t}^{\alpha }\) is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain \({\varOmega } \subset \mathbb {R}^{d}\) for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of \({D}_{t}^{\alpha }\) on a graded temporal mesh. The numerical method computes approximations \({u_{h}^{n}}\) and \({{p}_{h}^{n}}\) of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on \(\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}\) and \(\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}\)) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → 1−. Error bounds on \(\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}\) and \(\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}\) are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → 1−.
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Funding
The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grant NSAF-U1930402. The research of Chaobao Huang is supported in part by the National Natural Science Foundation of China under grants 11801332 and 11971276.
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Huang, C., Stynes, M. α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation. Numer Algor 87, 1749–1766 (2021). https://doi.org/10.1007/s11075-020-01036-y
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DOI: https://doi.org/10.1007/s11075-020-01036-y
Keywords
- Time-fractional problem
- Weak singularity
- Mixed finite element method
- Discrete Gronwall inequality
- α-robust