Variational and numerical analysis of a 𝐐-tensor model for smectic-A liquid crystals
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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 using the 𝒞^0 interior penalty method. More specifically, optimal rates in the H^1 and L^2 norms are obtained for 𝐐, while optimal rates in a mesh-dependent norm and L^2 norm are obtained for u. Numerical experiments confirm the rates of convergence.
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