Variational and numerical analysis of a $\mathbf{Q}$-tensor model for smectic-A liquid crystals

TL;DR

This paper investigates an energy minimisation model for smectic-A liquid crystals, proving solution existence and providing error estimates for numerical discretisation, supported by numerical experiments confirming convergence rates.

Contribution

It establishes the existence of solutions for a coupled PDE system and derives optimal error estimates for a specific discretisation method in the decoupled case.

Findings

Existence of solutions to the minimisation problem proven.

Optimal error rates obtained for the discretised system.

Numerical experiments confirm theoretical convergence rates.

Abstract

We analyse an energy minimisation problem recently proposed for modelling smectic-A liquid crystals. The optimality conditions give a coupled nonlinear system of partial differential equations, with a second-order equation for the tensor-valued nematic order parameter $\mathbf{Q}$ and a fourth-order equation for the scalar-valued smectic density variation $u$. Our two main results are a proof of the existence of solutions to the minimisation problem, and the derivation of a priori error estimates for its discretisation of the decoupled case (i.e., $q=0$) using the $\mathcal{C}^0$ interior penalty method. More specifically, optimal rates in the $H^1$ and $L^2$ norms are obtained for $\mathbf{Q}$, while optimal rates in a mesh-dependent norm and $L^2$ norm are obtained for $u$. Numerical experiments confirm the rates of convergence.