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 a 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 (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 emphasise 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. We figure out the
reason from the viewpoint of differential algebraic equations
and then remove the non-physical oscillation by doing only one
step of a modified backward Euler scheme at the initial time.
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. The talk is based
on a few joint papers with Chun Liu, Qi Wang, Xingbin Pan and
Roland Glowinski, etc.