Variational and numerical analysis of aQ-tensor model for smectic-A liquid crystals

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 parameterQand a fourth-order equation for the scalar-valued smectic density variationu. Our two main results are a proof of the existence of solutions to the minimisation problem, and the derivation ofa priorierror estimates for its discretisation of the decoupled case (i.e.,q= 0) using theC0interior penalty method. More specifically, optimal rates in theH1andL2norms are obtained forQ, while optimal rates in a mesh-dependent norm andL2norm are obtained foru. Numerical experiments confirm the rates of convergence.