Well-posedness and time stepping adaptivity for a class of collocation discretisations of time-fractional subdiffusion equations

Time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are discretised in time using collocation methods, which assume that the Caputo derivative of the computed solution is piecewise-polynomial. For such discretisations of any order $m\ge 0$, with any choice of collocation points, we give sufficient conditions for existence and uniqueness of collocation solutions. Furthermore, we investigate the applicability and performance of such schemes in the context of the a-posteriori error estimation and adaptive time stepping algorithms.