Daniel Kessler
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA
Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA
Alfred Schmidt
Zentrum fuer Technomathematik
Universitaet Bremen
28359 Bremen, Germany
Phase-field models, the simplest of which is Allen-Cahn's problem, are characterized by a small parameter $\eps$ that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on $\eps^{-2}$. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in $\eps^{-1}$. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.