Quasi-optimal and robust a posteriori error estimates in 𝐿∞(𝐿²) for the approximation of Allen-Cahn equations past singularities

Quasi-optimal a posteriori error estimates in L ∞ ( 0 , T ; L 2 ( Ω ) ) L^∞ (0,T;L^2(Ω )) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.