Non-uniform $\alpha$-Robust Alikhanov Mixed FEM with Optimal Convergence for the Time-Fractional Allen--Cahn Equation

TL;DR

This paper develops a mixed finite element method with a nonuniform Alikhanov scheme for the time-fractional Allen--Cahn equation, achieving optimal convergence and robustness with respect to the fractional order.

Contribution

It introduces a novel non-uniform Alikhanov scheme combined with mixed FEM for better accuracy and robustness in solving the time-fractional Allen--Cahn equation.

Findings

Achieves optimal $L^2$-error estimates for solution and flux.

Estimates are robust as fractional order $oldsymbol{ extit{ extalpha}}$ approaches 1.

Numerical experiments confirm theoretical convergence rates.

Abstract

We investigate a mixed finite element method for the spatial discretization of a time-fractional Allen--Cahn equation defined on a convex polyhedral domain, combined with a nonuniform Alikhanov scheme for the temporal approximation. Under suitable regularity assumptions on the initial data that are weaker than those typically imposed in the literature, we establish regularity results for the solution and its flux. We then derive optimal $L^2$-error estimates, up to a logarithmic factor, for both the solution and the flux. The estimates are robust with respect to the fractional order $\alpha$, in the sense that the associated constants remain bounded as $\alpha \to 1^{-}$. Numerical experiments are presented to confirm the theoretical findings.