Non-uniform $α$-Robust Alikhanov Mixed FEM with Optimal Convergence for the Time-Fractional Allen--Cahn Equation
This work provides an incremental improvement in numerical methods for fractional PDEs, benefiting researchers in computational mathematics.
The paper tackles the time-fractional Allen-Cahn equation by developing a mixed finite element method with a nonuniform temporal scheme, achieving optimal L^2-error estimates robust to the fractional order, as confirmed by numerical experiments.
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 $α$, in the sense that the associated constants remain bounded as $α\to 1^{-}$. Numerical experiments are presented to confirm the theoretical findings.