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Analysis of finite element approximations of a phase field model for two-phase fluids

Authors: Xiaobing Feng, Yinnian He and Chun Liu
Journal: Math. Comp. 76 (2007), 539-571
MSC (2000): Primary 65M60, 35K55, 76D05
Published electronically: November 20, 2006
MathSciNet review: 2291827
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Abstract: This paper studies a phase field model for the mixture of two immiscible and incompressible fluids. The model is described by a nonlinear parabolic system consisting of the nonstationary Stokes equations coupled with the Allen-Cahn equation through an extra phase induced stress term in the Stokes equations and a fluid induced transport term in the Allen-Cahn equation. Both semi-discrete and fully discrete finite element methods are developed for approximating the parabolic system. It is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model. Error estimates are derived for the semi-discrete method, and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.

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Additional Information

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

Yinnian He
Affiliation: Faculty of Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China

Chun Liu
Affiliation: Department of Mathematics, Penn State University, University Park, Pennsylvania 16802

Keywords: Two phase fluids, phase field model, Allen-Cahn equation, Stokes equations, finite element methods
Received by editor(s): April 28, 2005
Received by editor(s) in revised form: August 10, 2005
Published electronically: November 20, 2006
Additional Notes: The work of the first author was partially supported by the NSF grant DMS-0410266.
The work of the second author was partially supported by the NSF of China grant #10671154
Article copyright: © Copyright 2006 American Mathematical Society