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Quasi-optimal and robust a posteriori error estimates in $L^\infty (L^2)$ for the approximation of Allen-Cahn equations past singularities

Authors: Sören Bartels and Rüdiger Müller
Journal: Math. Comp. 80 (2011), 761-780
MSC (2010): Primary 65M15, 65M50; Secondary 35K20
Published electronically: November 18, 2010
MathSciNet review: 2772095
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Abstract: Quasi-optimal a posteriori error estimates in $L^\infty (0,T;L^2(\Omega ))$ are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.

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Additional Information

Sören Bartels
Affiliation: Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstrasse 6, 53115 Bonn, Germany

Rüdiger Müller
Affiliation: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Keywords: Allen-Cahn equation, mean curvature flow, finite element method, error analysis, adaptive methods
Received by editor(s): May 13, 2009
Received by editor(s) in revised form: February 1, 2010
Published electronically: November 18, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.