A Sharp $\alpha$-Robust $L^\infty (H^1)$ Error Bound for a Time-Fractional Allen-Cahn Problem Discretised by the Alikhanov $L2-1_\sigma$ Scheme and a Standard FEM

Abstract

A time-fractional Allen-Cahn initial-boundary value problem is considered, where the bounded spatial domain $\Omega$ lies in $\mathbb {R}^d$ for some $d \in \{1,2,3\}$ and has smooth boundary or is convex. A new a priori bound on certain derivatives of the unknown solution is derived. The problem is discretised in time by Alikhanov’s $L2-1_\sigma$ scheme on a graded mesh, while in space a standard finite element method is used. A new discrete fractional Gronwall inequality is proved that extends a previous discrete inequality; it is needed to handle a troublesome term in the error analysis. The computed solution is shown to attain the optimal convergence rate in $L^\infty (H^1(\Omega ))$; moreover, this error bound is $\alpha$-robust, where $\alpha \in (0,1)$ is the order of the temporal fractional derivative in the Allen-Cahn equation. Numerical experiments support the theoretical results.