Finite element analysis of a nematic liquid crystal Landau-de Gennes model with quartic elastic terms

TL;DR

This paper develops a finite element method for a $Q$-tensor liquid crystal model with quartic elastic terms, proving convergence and demonstrating simulations of phase transitions.

Contribution

It introduces an energy stable finite element scheme for a nematic liquid crystal Landau-de Gennes model with quartic elastic terms, including convergence analysis.

Findings

Successful simulation of isotropic-to-nematic phase transitions

Proof of scheme convergence via fixed-point iteration

Rigorous $ ext{Γ}$-convergence of discrete minimizers

Abstract

In arXiv:1906.09232v2, Golovaty et al. present a $Q$-tensor model for liquid crystal dynamics which reduces to the well-known Oseen-Frank director field model in uniaxial states. We study a closely related model and present an energy stable scheme for the corresponding gradient flow. We prove the convergence of this scheme via fixed-point iteration and rigorously show the $\Gamma$-convergence of discrete minimizers as the mesh size approaches zero. In the numerical experiments, we successfully simulate isotropic-to-nematic phase transitions as expected.