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