An Energy-Dissipation Finite Element Pressure-Correction Scheme for the Hydrodynamics of Smectic-A Liquid Crystals

In this paper, we develop a new linear, fully-decoupled, unconditional energy-stable
BDF2-SAV-FEM scheme for solving the smectic-A liquid crystals, based on the finite element
method (FEM) for spatial discretization and two-step backward differentiation formula (BDF2)
for temporal discretization. To decouple the computations of the layer function and velocity
field, we introduce an additional stabilization term into the constitutive equation. The nonlinear
energy potential and the Navier-Stokes equations are treated by the scalar auxiliary variable
(SAV) method and the rotational pressure-correction method, respectively. The unique solvability,
unconditional energy stability, and error estimations of the proposed numerical scheme have been
demonstrated. Several numerical experiments are carried out to validate our theoretical analysis.