Relaxation Methods for Liquid Crystal Problems

Abstract

A point relaxation method for the liquid crystal problem is proposed and analyzed. The liquid crystal problem is a minimization problem with a nonconvex local constraint. It is proved that the energy of successive iterates is nonincreasing for the point relaxation method with relaxation parameter $\omega $ satisfying $0 < \omega < 2$ for a class of material constants. It is also shown that the difference between successive iterates converges to zero, and that limit points of the iteration sequence are minima with respect to perturbations which are supported at a point and which satisfy the constraint. Numerical results are given which demonstrate the improved rate of convergence for overrelaxation.