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

top
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

top

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

top
  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
  4. [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. Zbl0811.35098
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.