Journal of Scientific Computing
Compactness Properties of the DG and CG Time Stepping Schemes for Parabolic Equations
SIAM Journal on Numerical Analysis
Journal of Computational Physics
On linear schemes for a Cahn-Hilliard diffuse interface model
Journal of Computational Physics
Journal of Computational Physics
Efficient numerical solution of discrete multi-component Cahn-Hilliard systems
Computers & Mathematics with Applications
Local enrichment of finite elements for interface problems
Computers and Structures
Time integration for diffuse interface models for two-phase flow
Journal of Computational Physics
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This paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the Navier--Stokes equations and the Cahn--Hilliard equation, which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids. In the model the two sets of equations are coupled through an extra phase induced stress term in the Navier--Stokes equations and a fluid induced transport term in the Cahn--Hilliard equation. Fully discrete mixed finite element methods are proposed for approximating the coupled system, it is shown that the proposed numerical methods satisfy a mass conservation law, and a discrete energy law which is analogous to the basic energy law for the phase field model. The convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law. As a by-product, the convergence result also provides a constructive proof of the existence of weak solutions to the Navier--Stokes-Cahn--Hilliard phase field model. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.