Mixed Methods for the Approximation of Liquid Crystal Flows

Abstract

Abstract. The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen-Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H 2 (Ω) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.Mathematics Subject Classification. 65M60, 76A15.

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

“…Direct application of their ideas results in numerical schemes which require either Hermite elements or a mixed method [28,29]. Subsequently there have been many developments and refinements.…”

“…The numerical approximation of solutions to the Lin-Liu equations has been studied extensively [2,6,13,[27][28][29]. These papers consider approximations of the weak statement…”

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

“…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.…”

“…The method proposed by Walkington and Liu in [25,26] considered C 1 Hermite polynomial approximations for which the lumping technique cannot be used.…”