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

摘要

A time-fractional Allen-Cahn initial-boundary value problem is considered, where the bounded spatial domain Omega lies in R-d for some d is an element of {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- l(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-infinity(H-1(Omega)); moreover, this error bound is alpha-robust, where alpha is an element of (0, 1) is the order of the temporal fractional derivative in the Allen-Cahn equation. Numerical experiments support the theoretical results.