Error analysis of a backward L2 method on graded meshes for linear and semilinear subdiffusion equations

Time-fractional linear and semilinear parabolic equations with a Caputo time derivative of order $\alpha \in (0,1)$ are considered. Their solutions exhibit an initial singularity of type $t^{\sigma }$ for any fixed $\sigma \in (0,1)\cup (1,2)$. A backward L2-type discrete fractional-derivative operator of order $3-\alpha$ is considered on nonuniform temporal meshes. Sufficient conditions for inverse monotonicity of this operator areestablished. Hence, sharp pointwise-in-time error bounds are obtained on graded temporal meshes with arbitrary degree of grading. The theoretical findings are illustrated by numerical experiments.