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

TL;DR

This paper analyzes collocation discretizations for time-fractional subdiffusion equations, establishing conditions for solution existence and uniqueness, and exploring adaptive time-stepping methods based on a-posteriori error estimates.

Contribution

It provides the first comprehensive analysis of well-posedness for collocation schemes of any order and collocation points, and demonstrates their effectiveness in adaptive algorithms.

Findings

Sufficient conditions for existence and uniqueness of collocation solutions.

Validation of adaptive time-stepping algorithms using a-posteriori error estimates.

Applicability of collocation methods to time-fractional subdiffusion equations.

Abstract

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.