A posteriori error control for the Allen-Cahn problem : circumventing Gronwall's inequality
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 1, p. 129-142

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.

DOI : https://doi.org/10.1051/m2an:2004006
Classification:  65M15,  65M50,  65M60
Keywords: a posteriori error estimates, phase-field models, adaptive finite element method
@article{M2AN_2004__38_1_129_0,
author = {Kessler, Daniel and Nochetto, Ricardo H. and Schmidt, Alfred},
title = {A posteriori error control for the Allen-Cahn problem : circumventing Gronwall's inequality},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
pages = {129-142},
doi = {10.1051/m2an:2004006},
zbl = {1075.65117},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_1_129_0}
}

Kessler, Daniel; Nochetto, Ricardo H.; Schmidt, Alfred. A posteriori error control for the Allen-Cahn problem : circumventing Gronwall's inequality. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 1, pp. 129-142. doi : 10.1051/m2an:2004006. http://www.numdam.org/item/M2AN_2004__38_1_129_0/

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