2012-06-21 Energy-law Preserving Continuous Finite Element Methods for a Few Physical Problems

Speaker: Prof. Ping Lin

                      Division of Mathematics,  University of Dundee

Time: 4:00 p.m. , Thursday, June, 21st, 2012

Venue:  Lecture Hall 104, Science Building, Tsinghua University

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 the Allen-Cahn phase field model for multiphase flows if the orientation variable is replaced by a phase function. There are a few large or small parameters involved in the model. 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 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 under some choices of parameters. We shall 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 our scheme non-physical oscillation near the interface occurs. A number of numerical examples demonstrate the good performance of the method. At the end of the talk we will show how to apply the method to compute a superconductivity model, especially at the regime of Hc2 or beyond, and the Marangoni-Benard convection of a two-fluids system. The talk is based on a few joint papers with Zhenlin Guo, Chun Liu, Qi Wang, Xingbin Pan and Roland Glowinski, etc.