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.
@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}, volume = {38}, year = {2004}, pages = {129-142}, doi = {10.1051/m2an:2004006}, zbl = {1075.65117}, language = {en}, url = {http://dml.mathdoc.fr/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, Tome 38 (2004) pp. 129-142. doi : 10.1051/m2an:2004006. http://gdmltest.u-ga.fr/item/M2AN_2004__38_1_129_0/
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