A finite element method for nematic liquid crystals with variable degree of orientation

We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field nn and its degree of orientation s, where the pair (s; n) minimizes a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for n, which allows for line and plane defects to have finite energy. We present a structure preserving discretization of the liquid crystal energy with piecewise linear finite elements that can handle the degenerate elliptic part without regularization, and we show that it is consistent and stable. We prove convergence of discrete global minimizers to continuous ones as the mesh size goes to zero. We develop a quasi-gradient flow scheme for computing discrete equilibrium solutions and prove that it has a strictly monotone energy decreasing property. We present simulations in two and three dimensions to illustrate the method's ability to handle nontrivial defects. The following online video illustrates the role of numerical analysis for the simulation of liquid crystal phenomena: [Walker, Shawn. \Mathematical Modeling and Simulation of Nematic Liquid Crystals (A Montage)." YouTube video, 04:13. Posted March 6, 2016. http://www.youtube. com/watch?v=pWWw7 6cQ-U].