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Relaxation methods for liquid crystal problems. (English) Zbl 0685.65058

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

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)
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