Lin, San-Yih; Luskin, Mitchell Relaxation methods for liquid crystal problems. (English) Zbl 0685.65058 SIAM J. Numer. Anal. 26, No. 6, 1310-1324 (1989). The liquid crystal problem involves finding the function \(N_{ijk}\approx n(x_{ijk})\) where \(n(x_{ijk})\) is determined by minimizing the Oseen-Frank free energy density \({\mathcal W}(n)=\int_{\Omega}W(\Delta n,n)dx\) in the space \(\Omega\) occupied by the liquid crystal material. A point relaxation method is proposed for this problem with a nonconvex local constraint. It is proved that the energy of successive iterates is nonincreasing for the point relaxation method with the relaxation parameter value \(0<\omega <2\) for certain material constants, furthermore that the difference between successive iterates converges to zero and finally that limit points of the iteration sequence are minima with respect to perturbations which are supported at a point. Numerical experiments show that overrelaxation improves the convergence for the problem. Reviewer: V.Burjan Cited in 18 Documents MSC: 65K10 Numerical optimization and variational techniques 65F10 Iterative numerical methods for linear systems 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) Keywords:liquid crystal problem; Oseen-Frank free energy density; point relaxation method; Numerical experiments; overrelaxation; convergence PDFBibTeX XMLCite \textit{S.-Y. Lin} and \textit{M. Luskin}, SIAM J. Numer. Anal. 26, No. 6, 1310--1324 (1989; Zbl 0685.65058) Full Text: DOI Link