We consider a degenerate parabolic system which models
the evolution of nematic liquid crystal with variable degree of orientation.
The system
is a slight modification
to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of
Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120].
We prove the global existence
of weak solutions by passing to the limit in a regularized system.
Moreover, we
propose a practical fully discrete finite element method for this
regularized
system, and we establish the (subsequence) convergence
of this finite element approximation
to the
solution of the regularized system as the mesh parameters tend to zero; and
to a solution of the original degenerate parabolic system
when the
the mesh and regularization parameters
all approach zero.
Finally, numerical experiments are included
which show the
formation, annihilation and evolution of line singularities/defects
in such models.