Projection methods in the context of nematic crystal flow

Abstract We present a finite element method based on a predictor-corrector scheme for the numerical approximation of the Ericksen–Leslie equations, a model for nematic liquid crystal flow including a nonconvex unit-sphere constraint. As a predictor step, we propose a linear, semi-implicit finite element discretization that naturally induces nodal orthogonality between the approximate director field and its time derivative. Afterwards, an explicit projection onto the unit-sphere is applied at every node without increasing the energy. Discrete well-posedness results and energy laws are established. Conditional convergence of the approximate solutions to energy-variational solutions to the Ericksen–Leslie equations is shown for a time-step restriction. Computational studies indicate the efficiency of the proposed linearization and the improved accuracy resulting from the inclusion of a projection step in the algorithm.