Finite Element Approximations of the Ericksen–Leslie Model for Nematic Liquid Crystal Flow

Abstract

The Ericksen-Leslie model describes dynamics of low molar-mass nematic liquid crystals, where the spatiotemporal distribution of defects defining texture is represented by the director unit vector field d : Ω T → S 2 . It consists of the Navier-Stokes equations with an extra viscous stress tensor, and a convective harmonic map heat flow equation to govern the dynamics of the director field. Two fully discrete finite element methods, for a regularized system using the Ginzburg-Landau regularization and for the limiting Ericksen-Leslie model, are proposed, and wellposedness and related discrete energy laws are established. For the regularized model, unconditional convergence of finite element solutions towards weak solutions of the continuum model as well as convergence towards measure-valued solutions of the limiting Ericksen-Leslie model are verified when the mesh parameters and the regularization parameter successively tend to zero. Computational experiments are also presented to show the importance of balancing numerical and regularization parameters, to compare regularized and direct approaches, and to show numerically the finite-time formation, annihilation, and evolution of point defects.

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

“…Subsequently there have been many developments and refinements. Various mixed methods have been considered in [2,3,13]; spectral schemes have been developed in [6]; methods with estimates independent of the penalty parameter were developed in [3,27]; and "semi-implicit" schemes which only require the solution of a linear system at each time step are developed in [13]. All of these schemes for this simplified model utilize the first order implicit Euler stepping scheme to evolve the solution in time.…”

Finite element approximation of nematic liquid crystal flows using a saddle-point structure

“…[17,4]). This reformulation will allow us to obtain finite element approximations with an energy estimate that mimics the one at the continuous level.…”

“…The use of this kind of initial approximation is ill-posed for the limit problem. 1 On the other hand, this initial condition has been used as a test problem in some numerical articles based on GL penalty problem approximations (see [23,25,26,4,17]). We refer to the numerical experimentation section for more details.…”

“…In the frame proposed herein, we can approximate numerically both the limit problem and GL penalized versions using the same numerical approximation. Existing algorithms [23,25,26,4,17] could not take ǫ arbitrarily small, since the condition number of the matrix blows up with ǫ −2 . As a rule of thumb, ǫ = 0.05 was the limit value used in numerical experiments.…”

“…This penalization term is in fact the one used in [4] for the GL term. The method proposed by Walkington and Liu in [25,26] considered C 1 Hermite polynomial approximations for which the lumping technique cannot be used.…”