Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows

Abstract

In this paper, we study a conditional long-time stable fully discrete finite element scheme for a Ginzburg-Landau model for nematic liquid crystal flow. We also obtain its time asymptotic convergence (when number of time steps go to infinity, fixed time step and mesh size) towards a unique critical point of the elastic energy subject to the finite element subspace. Finally, we estimate some convergence rates towards this limit critical point. To prove convergence of the whole sequence, a Lojasiewicz type inequality is used. <br>&nbsp; &nbsp Moreover, we extend these results to other schemes given in [3] and [10].