| The study of Liquid Crystal flows have been concerned for a long time after being presented. Since the theoretical and experimental study grow quickly, many significant method are derived and applied widely.We study the numerical simulation of the liquid crystal system. Find u ?u(x,t),p = p(x,t),d = d(x,t),such thatwith initial and boundary conditions:where Ω R2 is bounded and T is a positive constant, u, d: fi x R+ → R2 are the velocity and director fields of a liquid crystal and p is the pressure.D(u) = j(Vn +(Vn)T) is the streching tensor, is a penalty function used to approximate the constraint |d| = 1 and is the gradient of the scalar valued function F(d) = The continuum theory of the above system was proposed by Ericksen in [3] and [4] Leslie in [5] to model the flow of nematic liquid crystals. Since then there has been a remarkable increase in liquid crystal research in both theoretical and experimental aspects.The Ericksen-Leslie system was derived from the macroscopic point of view and involve many 00000000terms betweeen the two vector fields. In [10], Lin introduced the simplified system(l.l) consistiny of a Navier-Stokes type of equation. This system retains some important properties of the original Ericksen-Leslie equations and at the same time is amenable to 0000000analysis . Moreover, system(l.l) admits the following energy flowwhere E = 1/2||u||2L2 .Using energy estimates, Lin and Liu[ll] were able to prove local existence of classical solutions and global exitence of the weak solution to the system with u e L2[0, T; H1 SI)] L∞[0, T; L2(Ω)], d 6 L2[0, T; H2(Ω)] n LΩ[0, T; .H1 (Ω)].In this paper, we prove that the fully discritization numerical approximations will converge to the solution of (1.1). We use a finite element approximation of the spatial domain and Grank-Nicolson formate for the time. Then we get the error estimate in the L2- norm.Keywords:Liquid crystal;Grank-Nicoson;Finite element approximation...
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