Projection Methods in the Context of Nematic Crystal Flow

TL;DR

This paper introduces a predictor-corrector finite element method for simulating nematic liquid crystal flow, effectively handling the non-convex unit-sphere constraint with improved accuracy and energy stability.

Contribution

It develops a novel linear finite element scheme with a projection step for nematic flow, comparing continuous and discontinuous discretizations, and proves convergence and energy laws.

Findings

The method is computationally efficient and stable.

Including a projection step improves accuracy.

Convergence to energy-variational solutions is established.

Abstract

We present a continuous and a discontinuous linear Finite Element method based on a predictor-corrector scheme for the numerical approximation of the Ericksen-Leslie equations, a model for nematic liquid crystal flow including a non-convex unit-sphere constraint. As predictor step we propose a linear semi-implicit Finite Element discretization which naturally offers a local orthogonality relation between the approximate director field and its time derivative. Afterwards an explicit discrete projection onto the unit-sphere constraint is applied without increasing the modeled energy. For the Finite Element approximation of the director field, we compare the usage of a discrete inner product, usually referred to as mass-lumping, for a globally continuous, piecewise linear discretization to a piecewise constant, discontinuous Galerkin approach. Discrete well-posedness results and energy laws are established. Conditional convergence of the approximate solutions to energy-variational solutions of the Ericksen-Leslie equations is shown for a time-step restriction, see Theorems 1 and 2. Computational studies indicate the efficiency of the proposed linearization and the improved accuracy by including a projection step in the algorithm.