A Posteriori Error Control For The Allen-Cahn Problem: Circumventing Gronwall'S Inequality

Phase-field models, the simplest of which is Allen-Cahn's problem, are characterized by a small parameter epsilon 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 epsilon(-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 epsilon(-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.