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

@inproceedings{Xia2021VariationalAN, title={Variational and numerical analysis of a \$\mathbf\{Q\}\$-tensor model for smectic-A liquid crystals}, author={Jingmin Xia and Patrick E. Farrell}, year={2021} }

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…

## References

SHOWING 1-10 OF 25 REFERENCES

Discontinuous Galerkin Finite Element Methods for the Landau-de Gennes Minimization Problem of Liquid Crystals

- Mathematics, Computer ScienceIMA Journal of Numerical Analysis
- 2020

The quadratic convergence of the Newton iterates is proved along with complementary numerical experiments to establish the existence and local uniqueness of the discrete solution of the nonlinear problem.

A STRUCTURE-PRESERVING FEM FOR THE UNIAXIALLY CONSTRAINED Q-TENSOR MODEL OF NEMATIC LIQUID CRYSTALS

- 2019

We consider the one-constant Landau de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field Q, which is constrained to be uniaxial: Q = s(n⊗n−d−1I) where n is a…

Finite Element Analysis of the Landau--de Gennes Minimization Problem for Liquid Crystals

- Mathematics
- 1998

This paper describes the Landau--de Gennes free-energy minimization problem for computing equilibrium configurations of the tensor order parameter field that characterizes the molecular orientational…

Calculus of variations and its application to liquid crystals

- Mathematics
- 2014

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals.Throughout the thesis we will be looking to minimise integral functionals of the form I…

Some mixed finite element methods for biharmonic equation

- Mathematics
- 2000

Abstract Some perturbed mixed finite element methods related to the reduced integration technique are considered for solving the biharmonic equation problem. On a rectangular mesh, a similar scheme…

On the boundary value problem of the biharmonic operator on domains with angular corners

- Mathematics
- 1980

The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type…

Introduction to Q-tensor theory

- Physics
- 2014

This paper aims to provide an introduction to a basic form of the Q-tensor approach to modelling liquid crystals, which has seen increased interest in recent years. The increase in interest in this…

hp-version interior penalty DGFEMs for the biharmonic equation

- Mathematics
- 2007

We construct hp-version interior penalty discontinuous Galerkin finite element methods (DGFEMs) for the biharmonic equation, including symmetric and nonsymmetric interior penalty discontinuous…

A mixed method for 4th order problems using linear finite elements

- Computer Science
- 1978

It is shown that also in the case of linear finite éléments the mixed method approximations are convergent, and an error estimate in the L2 as well as in theL norm is proved.

From molecular to continuum modelling of bistable liquid crystal devices

- Materials Science
- 2017

ABSTRACT We study nematic equilibria on a square with tangent Dirichlet conditions on the edges, in three different modelling frameworks: (i) the off-lattice Hard Gaussian Overlap and Gay–Berne…