Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional $H^1$-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accuracy). To recover the theoretical accuracy in time, we propose an improved discrete Grönwall inequality and apply it to the well-known L1 formula and a fractional Crank–Nicolson scheme. With the help of a time-space error-splitting technique and the global consistency analysis, sharp $H^1$-norm error estimates of the two nonuniform approaches are established for a reaction–subdiffusion problems. Numerical experiments are included to confirm the sharpness of our analysis.

由于数值的Caputo导数具有解的固有初始奇异性和离散卷积形式，因此传统的 $H^{\mathrm{1个}}$分数次扩散问题的时间近似值的-范数分析（对应于经典扩散方程的情况）始终导致次优误差估计（时间准确性的损失）。为了及时恢复理论上的准确性，我们提出了一个改进的离散Grönwall不等式，并将其应用于著名的L1公式和分数Crank-Nicolson方案。借助时空错误拆分技术和全局一致性分析，锐利$H^{\mathrm{1个}}$针对反应-扩散问题建立了两种非均匀方法的-范数误差估计。包括数值实验，以确认我们分析的清晰度。