A linear mixed finite element scheme for a nematic Eriksen-Leslie liquid crystal model

In this talk we propose and analyze a fully discrete mixed scheme, by using continuous finite elements in space and a coupled linearized Euler type scheme in time, for solving a nematic liquid crystal model of the Eriksen-Leslie type by means of a penalized problem of the Ginzburg-Landau type. Conditional stability of this scheme is proved by means of a discrete version of the energy law established by Lin and Liu in [2]. We highlight our compactness results for the discrete velocity to prove the conditional convergence towards measure-valued solutions of the limiting Eriksen-Leslie problem (obtained by Becker, Feng, and Prohl) when the discrete parameters and the penalty parameter tend to zero at the same time. Finally, we test our finite element method with the numerical experience of annihilations of singularities presented in [4], and compare our results with those in [3] and [1].