An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative operator of order $3-\alpha$ is considered on nonuniform temporal meshes. Sufficient conditions for the inverse-monotonicity of this operator are established, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semi-discretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.

中文翻译：

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分数阶抛物线问题的分级网格L2型方法的误差分析

考虑具有分数阶 $\alpha\in(0,1)$ 的 Caputo 时间导数的初始边界值问题，其解通常在初始时间表现出奇异行为。在非均匀时间网格上考虑了阶 $3-\alpha$ 的 L2 型离散分数微分算子。建立了该算子的逆单调性的充分条件，这在具有任意分级程度的准分级时间网格上产生了尖锐的时间点误差界限。特别是，这些结果意味着更温和（与最佳相比）的分级在正时间内产生最佳收敛率。解决了时间上的半离散化和完全离散化。数值实验说明了理论发现。