偏微分方程讨论班
ECNU PDE Seminar

报告题目: Energy-law Preserving C^0 Finite Element Methods for Simulation of Liquid Crystal and Multi-phase Flows
演讲人： Professor Lin Ping
University of Dundee, UK
时间： 2009年6月5日（星期五）下午2：40-3：40
地点： 华东师大闵行校区数学系102室

Abstract
The liquid crystal (LC) flow model is a coupling between orientation (director field) of LC molecules and a flow field. The model may probably be one of simplest complex fluids and is very similar to an Allen-Cahn phase field model of multiphase flows if the orientation variable is replaced by a phase function. It is very much of multiscale feature due to a few large or small parameters involved in the model (e.g. the small penalty parameter for the unit length LC molecule or the small phase-change parameter, possibly large Reynolds number of the flow field, etc.). We propose a C^0 finite element formulation in space and a modified midpoint scheme in time which accurately preserves the inherent energy law of the model. We use C^0 elements because they are simpler than existing C^1 element and mixed element methods. We emphasize the energy law preservation because from the PDE analysis point of view the energy law is very important to correctly catch the evolution of singularities in the LC molecule orientation. In addition we will see numerical examples that the energy law preserving scheme performs better in the multiscale situation, for example, it works better for high Reynolds number flow fields and allows coarser grids to qualitatively catch the interface evolution of the two-phase flow. Finally we apply the same idea to a Cahn-Hilliard phase field model where the biharmonic operator is decomposed into two Laplacian operators. But we find that under a C^0 finite element setting non-physical oscillation near the interface occurs. From the viewpoint of differential algebraic equations this is because the modified midpoint scheme is not of stiff decay. The oscillation can be removed by doing only one step of a modified backward Euler scheme at the beginning time. A number of numerical examples demonstrate the case. We will mention possible application of the method to the superconductivity model as well.