Geodesic
A posteriori error control for the Allen-Cahn problem : circumventing Gronwall's inequality
Kessler, Daniel1 ; Nochetto, Ricardo H.  ; Schmidt, Alfred2
1 University of Maryland Department of Mathematics College Park MD 20740 USA
2 Zentrum für Technomathematik, Universität Bremen, Bibliothekstrasse 1, 28359 Bremen, Germany.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 38 (2004) no. 1, pp. 129-142

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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.

DOI : 10.1051/m2an:2004006
Classification : 65M15, 65M50, 65M60
Keywords: a posteriori error estimates, phase-field models, adaptive finite element method

Kessler, Daniel 1 ; Nochetto, Ricardo H.  ; Schmidt, Alfred 2

1 University of Maryland Department of Mathematics College Park MD 20740 USA
2 Zentrum für Technomathematik, Universität Bremen, Bibliothekstrasse 1, 28359 Bremen, Germany. 
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     title = {A posteriori error control for the {Allen-Cahn} problem : circumventing {Gronwall's} inequality},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
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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 , Tome 38 (2004) no. 1, pp. 129-142. doi: 10.1051/m2an:2004006

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