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-infinity(0, T; L-2(Omega)) 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.