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

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.