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Analysis of finite element approximations of a phase field model for two-phase fluids
Authors:Xiaobing Feng  Yinnian He  Chun Liu
Institution:Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996 ; Faculty of Science, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, People's Republic of China ; Department of Mathematics, Penn State University, University Park, Pennsylvania 16802
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.

Keywords:Two phase fluids  phase field model  Allen-Cahn equation  Stokes equations  finite element methods
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