For time-fractional parabolic equations with a Caputo time derivative of order $\alpha \in (0,1)$, we give pointwise-in-time a posteriori error bounds in the spatial $L_2$ and $L_{\infty }$ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates $2-\alpha$ in the presence of solution singularities.

中文翻译：

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时间分数抛物线方程的时间点后验误差控制

对于阶次为 Caputo 时间导数的时间分数抛物线方程 $\alpha \in (0,1)$，我们在空间上给出了时间点的后验误差界限 ${升}_2$ 和 ${升}_{\infty }$规范。因此，对 L1 方法应用了自适应网格构建算法，该算法可产生最佳收敛速度$2-\alpha$ 在存在解奇点的情况下。