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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  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
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