  A semilinear 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 L1-type discretizations of
this problem, we employ the method of upper and lower solutions to obtain 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 the optimal convergence rate $2-\alpha$ in positive
time, while quasi-uniform temporal meshes yield first-order convergence in
positive time. Furthermore, under appropriate conditions on the nonlinearity,
the method of upper and lower solutions immediately implies that, similarly to
the exact solutions, the computed solutions lie within a certain range.
Semi-discretizations in time and full discretizations using finite differences
and finite elements in space are addressed. The theoretical findings are
illustrated by numerical experiments.
