  In this paper we propose a time-splitting finite-element scheme for
approximating solutions of the Ericksen-Leslie equations governing the flow of
nematic liquid crystals. These equations are to be solved for a velocity vector
field and a scalar pressure as well as a director vector field representing the
direction along which the molecules of the liquid crystal are oriented.
  The algorithm is designed at two levels. First, at the variational level, the
velocity, pressure and director are computed separately, but the director field
has to be computed together with an auxiliary variable in order to deduce a
priori energy estimates. Second, at the algebraic level, one can avoid
computing such an auxiliary variable if this is approximated by a piecewise
constant finite-element space. Therefore, these two steps give rise to a
numerical algorithm that computes separately only the primary variables.
Moreover, we will use a pressure stabilization technique that allows an stable
equal-order interpolation for the velocity and the pressure. Finally, some
numerical simulations are performed in order to show the robustness and
efficiency of the proposed numerical scheme and its accuracy.
