A stabilized divergence-free virtual element scheme for the nematic liquid crystal flows

In this work, we propose and analyze a stabilized divergence-free virtual element method (VEM) for approximating the hydrodynamics system of nematic liquid crystal flows. By adding appropriate stabilization term so that the nonlinear potential function can be treated explicitly, and using the implicit-explicit (IMEX) approach to handle the nonlinear coupling terms, we develop a linear and energy-stable fully discrete virtual element scheme. We further prove that the fully discrete scheme is uniquely solvable and energy stable in discrete sense, moreover, we obtain the optimal error estimates rigorously. Finally, numerical examples are provided to demonstrate the accuracy, stability and efficiency of the proposed scheme.