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A semilinear initial-boundary value problem with a Caputo time derivative offractional order $\alpha\in(0,1)$ is considered, solutions of which typicallyexhibit a singular behaviour at an initial time. For L1-type discretizations ofthis problem, we employ the method of upper and lower solutions to obtain sharppointwise-in-time error bounds on quasi-graded temporal meshes with arbitrarydegree of grading. In particular, those results imply that milder (compared tothe optimal) grading yields the optimal convergence rate $2-\alpha$ in positivetime, while quasi-uniform temporal meshes yield first-order convergence inpositive time. Furthermore, under appropriate conditions on the nonlinearity,the method of upper and lower solutions immediately implies that, similarly tothe exact solutions, the computed solutions lie within a certain range.Semi-discretizations in time and full discretizations using finite differencesand finite elements in space are addressed. The theoretical findings areillustrated by numerical experiments.

Error analysis for time-fractional semilinear parabolic equations using upper and lower solutions