Research On The Finite Element Algorithms Of Simplified Ericksen-Leslie Model

Ericksen-Leslie model is a very important model to describe the hydrodynamics of nematic liquid crystals.The complete form of Ericksen-Leslie model is very complex,so this artical mainly research the simplified Ericksen-Leslie model which ignores Leslie stress.Because the simplified Ericksen-Leslie model has the problems of incompressibility,strong nonlinearity,nonconvex constraint and coupling.It is very difficult for us to solve the model.Therefore,we propose two finite element algorithms for the simplified Ericksen-Leslie model:decoupling algorithm with different time steps and modular gra-div stabilized finite element algorithm.For the simplified Ericksen-Leslie model,we propose a decoupling algorithm with different time steps.The algorithm allows different time steps for different physical fields.A fully discrete decoupling approximation scheme with different time steps is constructed.The decoupling algorithm with different time steps can greatly improve the computational efficiency.We propose a modular grad-div stable finite element method for nematic liquid crystal flow,in which the grad-div stabilized term is added to the backward Euler scheme.Through numerical examples,we find that the numerical error of non modular grad-div stabilized finite element method increases with the decrease of viscosity coefficient,but the error of non modular grad-div stabilized finite element method increases with the decrease of viscosity coefficient.Our modular grad-div stabilized finite element method is still computable and stable.