no. 1
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: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 129-142

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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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

[1] S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085-1095.

[2] H. Brézis, Analyse fonctionnelle. Dunod, Paris (1999). | Zbl

[3] G. Caginalp and X. Chen, Convergence of the phase-field model to its sharp interface limits. Euro. J. Appl. Math. 9 (1998) 417-445. | Zbl

[4] X. Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differantial Equations 19 (1994) 1371-1395. | Zbl

[5] Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér 9 (1975) 77-84. | Zbl | Numdam

[6] R. Dautrey and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Masson (1988). | Zbl

[7] P. De Mottoni and M. Schatzman, Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533-1589. | Zbl

[8] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems iv: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729-1749. | Zbl

[9] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Num. Math. 94 (2003) 33-65. | Zbl

[10] Ch. Makridakis and R.H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41 (2003) 1585-1594. | Zbl

[11] J. Rappaz and J.-F. Scheid, Existence of solutions to a phase-field model for the solidification process of a binary alloy. Math. Methods Appl. Sci. 23 (2000) 491-513. | Zbl

[12] A. Schmidt and K. Siebert, ALBERT: An adaptive hierarchical finite element toolbox. Preprint 06/2000, Freiburg edition. | MR

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