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Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation. (English) Zbl 1097.35082
The main topic of paper is studying a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation of the type: $\partial_t s=k_1 \text{div}([1+\delta| \nabla s| ^{p-2}]\nabla s)- k_2 | \nabla \varphi| ^2 s -W^\prime (s)$ $s^2\partial_t \varphi =k_3\, \text{div}(s^2 \nabla \varphi)$ with appropriate initial condition and a Dirichlet boundary condition on some bounded maximum three dimensional space domain $$\Omega$$ and some time interval $$(0,T)$$. Here the unknown scalar function $$s$$ represents averaged local orientation of molecules compared to unit vector $$\mathbf{n} (x,t)$$ called director and only planar director configurations $$\mathbf{n} = (\cos \varphi, \sin \varphi)$$, where $$\varphi$$ is the associated angle is considered. $$k_i, i=1,2,3>0$$, $$\delta\geq0$$ are given real numbers and $$W$$ is a smooth double well potential. It is the modification of previous models investigated by Calderer, Golovaty, Lin and Liu with $$\delta=0$$ and is a special case of Erickesen’s general continuum model. A regularized system with regularization parameter in second equation is proposed and well posedness of this system is proved. Moreover the existence of a weak solution and a certain dissipative energy law is proved. Then passing to the limit for regularization parameter, the same is proved also for the weak solution of the original problem. Practical fully discrete finite element method is then derived to obtain numerical solutions of the proposed problem, and a convergence of an approximation solution to the continuous one is presented in some functional spaces. Several numerical experiments for two space dimension domain are included. On their results it is shown the formation, annihilation and evolution of line singularities in such models.

MSC:
 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations 35Q35 PDEs in connection with fluid mechanics 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 76A15 Liquid crystals 35K50 Systems of parabolic equations, boundary value problems (MSC2000)
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