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

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.