Optimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen--Cahn equations past singularities
Bartels, Sören, Müller, Rüdiger · Weierstraß-Institut für Angewandte Analysis und Stochastik · 2009
Optimal a posteriori error estimates in $L^infty(0,T;L^2(O))$ 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.