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

Daniel Kessler[1]; Ricardo H. Nochetto; Alfred Schmidt[2]

  • [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 - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 1, page 129-142
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Kessler, Daniel, Nochetto, Ricardo H., and 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 38.1 (2004): 129-142. <http://eudml.org/doc/245771>.

@article{Kessler2004,
abstract = {Phase-field models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter $\{\varepsilon \}$ 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 $\{\varepsilon \}^\{-2\}$. 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 $\{\varepsilon \}^\{-1\}$. 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.},
affiliation = {University of Maryland Department of Mathematics College Park MD 20740 USA; Zentrum für Technomathematik, Universität Bremen, Bibliothekstrasse 1, 28359 Bremen, Germany.},
author = {Kessler, Daniel, Nochetto, Ricardo H., Schmidt, Alfred},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {a posteriori error estimates; phase-field models; adaptive finite element method; Allen-Cahn equation; Phase-field models; mesh adaptation techniques; a posteriori error control; adaptive finite element},
language = {eng},
number = {1},
pages = {129-142},
publisher = {EDP-Sciences},
title = {A posteriori error control for the Allen–Cahn problem : circumventing Gronwall’s inequality},
url = {http://eudml.org/doc/245771},
volume = {38},
year = {2004},
}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 1
SP - 129
EP - 142
AB - Phase-field models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter ${\varepsilon }$ 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 ${\varepsilon }^{-2}$. 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 ${\varepsilon }^{-1}$. 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.
LA - eng
KW - a posteriori error estimates; phase-field models; adaptive finite element method; Allen-Cahn equation; Phase-field models; mesh adaptation techniques; a posteriori error control; adaptive finite element
UR - http://eudml.org/doc/245771
ER -

References

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  1. [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. [2] H. Brézis, Analyse fonctionnelle. Dunod, Paris (1999). Zbl0511.46001
  3. [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. Zbl0930.35024
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  5. [5] Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér 9 (1975) 77–84. Zbl0368.65008
  6. [6] R. Dautrey and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Masson (1988). Zbl0642.35001
  7. [7] P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533–1589. Zbl0840.35010
  8. [8] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems iv: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749. Zbl0835.65116
  9. [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. Zbl1029.65093
  10. [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. Zbl1052.65088
  11. [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. Zbl0964.35026
  12. [12] A. Schmidt and K. Siebert, ALBERT: An adaptive hierarchical finite element toolbox. Preprint 06/2000, Freiburg edition. MR1784069

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