Numerous error estimates have been carried out on various numerical schemes for subdiffusion equations. Unfortunately most error bounds suffer from a factor $1/(1-\alpha)$ or $\Gamma(1-\alpha)$, which blows up as the fractional order $\alpha\to 1^-$, a phenomenon not consistent with regularity of the continuous problem and numerical simulations in practice. Although efforts have been made to avoid the factor blow-up phenomenon, a robust analysis of error estimates still remains incomplete for numerical schemes with general nonuniform time steps. In this paper, we will consider the $\alpha$-robust error analysis of convolution-type schemes for subdiffusion equations with general nonuniform time-steps, and provide explicit factors in error bounds with dependence information on $\alpha$ and temporal mesh sizes. As illustration, we apply our abstract framework to two widely used schemes, i.e., the L1 scheme and Alikhanov's scheme. Our rigorous proofs reveal that the stability and convergence of a class of convolution-type schemes is $\alpha$-robust, i.e., the factor will not blowup while $\alpha\to 1^-$ with general nonuniform time steps even when rather general initial regularity condition is considered.

翻译：针对次扩散方程的各种数值格式已开展了大量误差估计研究。然而，大多数误差界包含因子 $1/(1-\alpha)$ 或 $\Gamma(1-\alpha)$，当分数阶 $\alpha\to 1^-$ 时该因子发散，这一现象与连续问题的正则性及实际数值模拟结果不一致。尽管已有研究试图避免因子发散现象，但对于一般非均匀时间步长的数值格式，其误差估计的鲁棒性分析仍不完整。本文针对具有一般非均匀时间步长的次扩散方程，研究卷积型格式的 $\alpha$-鲁棒误差分析，并给出显式误差界因子，其中包含 $\alpha$ 和时间网格尺寸的依赖信息。作为示例，我们将抽象框架应用于两种广泛使用的格式，即L1格式和Alikhanov格式。严格证明揭示了一类卷积型格式的稳定性和收敛性具有 $\alpha$-鲁棒性，即在一般非均匀时间步长下，即使考虑相当一般的初始正则性条件，当 $\alpha\to 1^-$ 时因子也不会发散。