(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
In this paper, we propose two first-order, unconditionally stable, fully decoupled methods based on the Gauge-Uzawa method for solving the penalty Ericksen-Leslie system with variable density. The first method is based on the scalar auxiliary variable method for explicitly handling the nonlinear part, and the Gauge-Uzawa method for decoupling the velocity and pressure. The second method is formulated by introducing an auxiliary intermediate velocity variable for decoupling the director field computation from the velocity field, and then combining it with the Gauge-Uzawa method and the convex splitting method. Thus, two highly efficient, fully decoupled schemes are built. Furthermore, we use the finite element method to give an efficient implementation of the schemes. Then, we prove that both two schemes are unconditionally stable. Finally, numerical examples are provided to verify the convergence rate, unconditional stability, and computational efficiency. The proposed methods also provide a solution for establishing numerical schemes for other highly nonlinear variable-density coupled systems.