# Discontinuous Galerkin Finite Element Methods for the Landau-de Gennes Minimization Problem of Liquid Crystals

## TL;DR

This paper develops and analyzes discontinuous Galerkin finite element methods for modeling equilibrium states of liquid crystals, providing stability, error estimates, and convergence results for the nonlinear Landau-de Gennes minimization problem.

## Contribution

It introduces a stable discontinuous Galerkin approach for the nonlinear Landau-de Gennes model, with rigorous error analysis and convergence proofs.

## Key findings

- Stable discretization of the nonlinear problem
- Optimal a priori error estimates in energy norm
- Quadratic convergence of Newton's method

## Abstract

We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element methods are used to approximate the solutions of this nonlinear problem with non-homogeneous Dirichlet boundary conditions. A discrete inf-sup condition demonstrates the stability of the discontinuous Galerkin discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the non-linear problem. An a priori error estimate in the energy norm is derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of Newton's iterates along with complementary numerical experiments.