  A numerical method is developed for solving a system of partial differential
equations modeling the flow of a nematic liquid crystal fluid with stretching
effect, which takes into account the geometrical shape of its molecules. This
system couples the velocity vector, the scalar pressure and the director vector
representing the direction along which the molecules are oriented. The scheme
is designed by using finite elements in space and a time-splitting algorithm to
uncouple the calculation of the variables: the velocity and pressure are
computed by using a projection-based algorithm and the director is computed
jointly to an auxiliary variable. Moreover, the computation of this auxiliary
variable can be avoided at the discrete level by using piecewise constant
finite elements in its approximation. Finally, we use a pressure stabilization
technique allowing a stable equal-order interpolation for the velocity and the
pressure. Numerical experiments concerning annihilation of singularities are
presented to show the stability and efficiency of the scheme.
