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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 dictatesthe 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.

Information

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

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