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A time-splitting finite-element approximation for the Ericksen-Leslie equations

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.