Gamma-convergent projection-free finite element methods for nematic liquid crystals: The Ericksen model

TL;DR

This paper introduces a simple, projection-free finite element method for the Ericksen model of nematic liquid crystals, capable of accurately capturing defects and ensuring stability and convergence in 2D and 3D simulations.

Contribution

The paper presents a novel finite element scheme that is projection-free, easy to implement, and proven to be stable and $ ext{Gamma}$-convergent for defect modeling in nematic liquid crystals.

Findings

Effective in capturing complex defects in 2D and 3D

Proven stability and $ ext{Gamma}$-convergence of the method

Demonstrated ease of implementation within standard finite element packages

Abstract

The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable, and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages. Our scheme is projection-free and thus circumvents the use of weakly acute meshes, which are quite restrictive in 3D but are required by recent algorithms for convergence. We prove stability and $\Gamma$-convergence properties of the new method in the presence of defects. We also design an effective nested gradient flow algorithm for computing minimizers that controls the violation of the unit-length constraint of the director. We present several simulations in 2D and 3D that document the performance of the proposed scheme and its ability to capture quite intriguing defects.