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.