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An essential feature of the subdiffusion equations with the $\alpha$-ordertime fractional derivative is the weak singularity at the initial time. Theweak regularity of the solution is usually characterized by a regularityparameter $\sigma\in (0,1)\cup(1,2)$. Under this general regularity assumption,we here obtain the pointwise-in-time error estimate of the widely used L1scheme for nonlinear subdiffusion equations. To the end, we present a refineddiscrete fractional-type Gr\"onwall inequality and a rigorous analysis for thetruncation errors. Numerical experiments are provided to demonstrate theeffectiveness of our theoretical analysis.

Sharp pointwise-in-time error estimate of L1 scheme for nonlinear subdiffusion equations