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.

翻译：对于具有Caputo时间衍生物($\alpha\ in,0,1,1美元)的定时抛物线方程,我们给出空间值为$L_2美元和$L ⁇ infty美元规范的事后误差的近似值。因此,对L1方法采用了适应性网状构建算法,在有解算奇数的情况下,该方法得出最佳趋同率为$2-alpha美元。