A posteriori error estimates for the allen-cahn problem

This work is concerned with the proof of a posteriori error estimates for fully discrete Galerkin approximations of the Allen--Cahn equation in two and three spatial dimensions. The numerical method comprises the backward Euler method combined with conforming finite elements in space. For this method, we prove conditional type a posteriori error estimates in the $L^{}_4(0,T;L^{}_4(\Omega))$-norm that depend polynomially upon the inverse of the interface length $\epsilon$. The derivation relies crucially on the availability of a spectral estimate for the linearized Allen--Cahn operator about the approximating solution in conjunction with a continuation argument and a variant of the elliptic reconstruction. The new analysis also appears to improve variants of known a posteriori error bounds in $L_2(H^1)$, $L_\infty^{}(L_2^{})$-norms in certain regimes.