Paper Info
Title
Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits
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 epsilon, 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 1/epsilon only in some lower polynomial order for small epsilon. 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.
Year
DOI
Venue
2004
10.1090/S0025-5718-03-01588-6
MATHEMATICS OF COMPUTATION
Keywords
Field
DocType
phase field model,Allen-Cahn equation,Cahn-Hilliard equation,Stefan problem,motion by mean curvature,Hele-Shaw model,fully discrete finite element method
Mean curvature flow,Polynomial,Mathematical analysis,Mean curvature,Finite element method,Numerical analysis,Partial differential equation,Stefan problem,Numerical stability,Mathematics
Journal
Volume
Issue
ISSN
73
246
0025-5718
Citations 
PageRank 
References 
15
9.90
2
Authors
2
Name
Order
Citations
PageRank
Xiaobing Feng1906112.55
Andreas Prohl230267.29