Abstract
A time-fractional Allen-Cahn problem is considered, where the spatial domain Ω is a bounded subset of \(\mathbb {R}^{d}\) for some d ∈{1,2,3}. New bounds on certain derivatives of the solution are derived. These are used in the analysis of a numerical method (L1 discretization of the temporal fractional derivative on a graded mesh, with a standard finite element discretization of the spatial diffusion term, and Newton linearization of the nonlinear driving term), showing that the computed solution achieves the optimal rate of convergence in the Sobolev H1(Ω) norm. (Previous papers considered only convergence in L2(Ω).) Numerical results confirm our theoretical findings.
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Acknowledgments
We thank an unknown reviewer for pointing out an error in the analysis of our original paper. This work was completed while Chaobao Huang was visiting Beijing CSRC.
Funding
The research of Chaobao Huang is supported in part by the National Natural Science Foundation of PR China (Grant Nos. 11801332 and 11971276). The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grant NSAF-U1930402.
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Communicated by: Lourenco Beirao da Veiga
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Huang, C., Stynes, M. Optimal H1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation. Adv Comput Math 46, 63 (2020). https://doi.org/10.1007/s10444-020-09805-y
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DOI: https://doi.org/10.1007/s10444-020-09805-y