Optimal and robust a posteriori error estimates in $L^infty(L^2)$ for the approximation of Allen--Cahn equations past singularities
- Bartels, Sören
- Müller, Rüdiger
2010 Mathematics Subject Classification
- 65M60 65M15 35K55
- Allen-Cahn equation, mean curvature flow, finite element method, error analysis, adaptive methods
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.
- Math. Comp., 80 (2011) pp. 761--780.