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A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality

Published online by Cambridge University Press:  15 February 2004

Daniel Kessler ,
Ricardo H. Nochetto  and
Alfred Schmidt
Daniel Kessler
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. rhn@math.umd.edu.
Ricardo H. Nochetto
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. rhn@math.umd.edu. Institute for Physical Sciences and Technology, College Park, MD 20742, USA.
Alfred Schmidt
Affiliation:
Zentrum für Technomathematik, Universität Bremen, Bibliothekstrasse 1, 28359 Bremen, Germany.
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Abstract

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.

Keywords

A posteriori error estimatesphase-field modelsadaptive finite element method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 1 , January 2004 , pp. 129 - 142
Copyright
© EDP Sciences, SMAI, 2004

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References

Allen, S.M. and Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085–1095. CrossRef
H. Brézis, Analyse fonctionnelle. Dunod, Paris (1999).
Caginalp, G. and Chen, X., Convergence of the phase-field model to its sharp interface limits. Euro. J. Appl. Math. 9 (1998) 417–445. CrossRef
Chen, X., Spectrum for the Allen–Cahn, Cahn–Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differantial Equations 19 (1994) 1371–1395. CrossRef
Clément, Ph., Approximation by finite element functions using local regularization. RAIRO Anal. Numér 9 (1975) 77–84.
R. Dautrey and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Masson (1988).
de Mottoni, P. and Schatzman, M., Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533–1589. CrossRef
Eriksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems iv: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749. CrossRef
Feng, X. and Prohl, A., Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Num. Math. 94 (2003) 33–65. CrossRef
Ch. Makridakis, R.H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41 (2003) 1585–1594. CrossRef
Rappaz, J. and Scheid, J.-F., Existence of solutions to a phase-field model for the solidification process of a binary alloy. Math. Methods Appl. Sci. 23 (2000) 491–513. 3.0.CO;2-4>CrossRef
A. Schmidt and K. Siebert, ALBERT: An adaptive hierarchical finite element toolbox. Preprint 06/2000, Freiburg edition.