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Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

Authors: Xiaobing Feng and Andreas Prohl
Journal: Math. Comp. 73 (2004), 541-567
MSC (2000): Primary 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99
Published electronically: July 28, 2003
MathSciNet review: 2028419
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Abstract: We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter $\varepsilon$, known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size $h$ and the time step size $k$. In particular, it is shown that all error bounds depend on $\frac{1}{\varepsilon}$ only in some lower polynomial order for small $\varepsilon$. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

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Additional Information

Xiaobing Feng
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996

Andreas Prohl
Affiliation: Department of Mathematics, ETH, CH-8092 Zürich, Switzerland

Keywords: Phase field model, Allen-Cahn equation, Cahn-Hilliard equation, Stefan problem, motion by mean curvature, Hele-Shaw model, fully discrete finite element method
Received by editor(s): November 16, 2001
Received by editor(s) in revised form: October 30, 2002
Published electronically: July 28, 2003
Article copyright: © Copyright 2003 American Mathematical Society