A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Grönwall inequality is employed to give an α-robust $H^1$-norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative $\alpha \to 1^{-}$. Numerical results show that the theoretical analysis is sharp.

中文翻译：

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二维时间分数扩散方程ADI方案的α- Robust H ^{1 -}范数收敛分析

针对具有弱奇异解的二维时间分数扩散方程，提出了一种完全离散的ADI方案，其中采用梯度网格上的L1方案来解决初始奇异性问题。改进的离散分数 Grönwall 不等式用于给出α-稳健$H^1$-完全离散ADI方案的范数收敛分析，其中在分数阶导数的阶数时误差界限不会爆炸 $\alpha \to 1^{-}$. 数值结果表明，理论分析是尖锐的。