  An initial-boundary value problem with a Caputo time derivative of fractional
order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a
singular behaviour at an initial time. An L2-type discrete
fractional-derivative operator of order $3-\alpha$ is considered on nonuniform
temporal meshes. Sufficient conditions for the inverse-monotonicity of this
operator are established, which yields sharp pointwise-in-time error bounds on
quasi-graded temporal meshes with arbitrary degree of grading. In particular,
those results imply that milder (compared to the optimal) grading yields
optimal convergence rates in positive time. Semi-discretizations in time and
full discretizations are addressed. The theoretical findings are illustrated by
numerical experiments.
