Date of Award


Document Type

Open Access Dissertation




College of Arts and Sciences

First Advisor

Xiaofeng Yang


The thesis consists of two parts. In the first part we propose several second order in time, fully discrete, linear and nonlinear numerical schemes to solve the phase-field model of two-phase incompressible flows in the framework of finite element method. The schemes are based on the second order Crank-Nicolson method for time disretizations, projection method for Navier-Stokes equations, as well as several implicit-explicit treatments for phase-field equations. The energy stability, solvability, and uniqueness for numerical solutions of proposed schemes are further proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes thereafter.

In the second part we consider the numerical approximations for the model of smectic-A liquid crystal flows. The model equation, that is derived from the variational approach of the de Gennes energy, is a highly nonlinear system that couples the incompressible Navier-Stokes equations and two nonlinear coupled second-order elliptic equations. Based on some subtle explicit-implicit treatments for nonlinear terms, we develop unconditionally energy stable, linear, decoupled time discretization scheme. We also rigorously prove that the proposed scheme obeys the energy dissipation law. Various numerical simulations are presented to demonstrate the accuracy and the stability thereafter.

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Mathematics Commons