A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model∗
1 Dpto. E.D.A.N., University of
Sevilla, Aptdo. 1160, 41080
2 Dpto. Matemática Aplicada I, University of Sevilla, Av. Reina Mercedes s/n, 41012 Sevilla, Spain. .
Revised: 22 February 2013
In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.
Mathematics Subject Classification: 35Q35 / 65M12 / 65M60
Key words: Liquid crystal / Navier–Stokes / stability / convergence / finite elements / penalization
© EDP Sciences, SMAI, 2013