Variational and numerical analysis of a 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 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 C0 interior penalty method. More specifically, optimal rates in the H1 and L2 norms are obtained for Q, while optimal rates in a mesh-dependent norm and L2 norm are obtained for u. Numerical experiments confirm the rates of convergence.