The sharp error estimate of a Grünwald–Letnikov (GL) scheme on uniform mesh for reaction-subdiffusion equations with weakly singular solutions is considered, where the spatial domain is a square in R² which is discretized by Legendre Galerkin spectral method. A discrete fractional Grönwall inequality is shown by constructing a family of discrete complementary convolution (DCC) kernels for the discrete convolution coefficients of GL scheme, which is used to get the stability and convergence of the fully discrete scheme. By a delicate analysis of the convolution sum of DCC kernels and truncation errors, we also show that the error estimate is sharp and α-robust as the fractional order \(\alpha \rightarrow 1^{-}\), which avoids the so-called coefficient blow-up phenomenon. Numerical examples are provided to verify the sharpness of our theoretical analysis.

考虑了具有弱奇异解的反应-亚扩散方程在均匀网格上的 Grünwald-Letnikov (GL) 方案的尖锐误差估计，其中空间域是R ^{2 中的}一个正方形，其由 Legendre Galerkin 谱方法离散。通过为 GL 方案的离散卷积系数构造一系列离散互补卷积 (DCC) 核，显示了离散分数 Grönwall 不等式，用于获得完全离散方案的稳定性和收敛性。通过对 DCC 核的卷积和和截断误差的精细分析，我们还表明误差估计是尖锐的和α稳健的作为分数阶\(\alpha \rightarrow 1^{-}\)，避免了所谓的系数膨胀现象。提供了数值例子来验证我们的理论分析的锐度。