  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. For this problem, we give a simple and
general numerical-stability analysis using barrier functions, which yields
sharp pointwise-in-time error bounds on quasi-graded temporal meshes with
arbitrary degree of grading. L1-type and Alikhanov-type discretization in time
are considered. 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.
