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.

中文翻译：

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时间分数阶Allen-Cahn方程的完全离散有限元方法的最优H 1空间收敛

一种时间分数的Allen-卡恩问题被认为是，在空间域Ω是有界集\（\ mathbb {R} ^ {d} \）一段d ∈{1,2,3}。得出该解的某些导数的新界。这些用于数值方法的分析（梯度网格上的时间分数导数的L1离散化，空间扩散项的标准有限元离散化和非线性驱动项的牛顿线性化），表明计算出的解决方案在Sobolev H ¹（Ω）范数中达到最佳收敛速度。（以前的论文只考虑了L ²（Ω）。）数值结果证实了我们的理论发现。