A Convergent Finite Element Scheme for the Q-Tensor Model of Liquid Crystals Subjected to an Electric Field

TL;DR

This paper introduces a stable, convergent finite element scheme for simulating the Landau-de Gennes Q-tensor model of liquid crystals under electric fields, enabling accurate numerical analysis of phenomena like the Fréedericksz transition.

Contribution

The paper develops a fully discrete, stable numerical scheme with convergence proof for the Q-tensor model under electric fields, incorporating a convex splitting and tensor truncation.

Findings

The scheme is stable and well-posed.

Solutions converge to weak solutions as discretization refines.

Numerical simulations successfully reproduce the Fréedericksz transition.

Abstract

We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which we show that it is possible to simulate the Fr\'eedericksz transition with this scheme.