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 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 . 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 . 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.
Mots-clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {129--142},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
doi = {10.1051/m2an:2004006},
zbl = {1075.65117},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004006/}
}
TY - JOUR AU - Kessler, Daniel AU - Nochetto, Ricardo H. AU - Schmidt, Alfred TI - A posteriori error control for the Allen-Cahn problem : circumventing Gronwall's inequality JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 129 EP - 142 VL - 38 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004006/ DO - 10.1051/m2an:2004006 LA - en ID - M2AN_2004__38_1_129_0 ER -
%0 Journal Article %A Kessler, Daniel %A Nochetto, Ricardo H. %A Schmidt, Alfred %T A posteriori error control for the Allen-Cahn problem : circumventing Gronwall's inequality %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 129-142 %V 38 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004006/ %R 10.1051/m2an:2004006 %G en %F 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: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 129-142. doi : 10.1051/m2an:2004006. http://www.numdam.org/articles/10.1051/m2an:2004006/
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