error control for the Allen–Cahn problem: circumventing Gronwall's inequality
Daniel Kessler, Ricardo H. Nochetto, Alfred Schmidt (2010)
ESAIM: Mathematical Modelling and Numerical Analysis
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Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on 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. Using an energy argument combined with a topological continuation argument and a...