A Linearized, Decoupled and Unconditionally Stable BDF2 FEM for the Active Fluid Model

In this paper, we develop a linear and decoupled fully discrete mixed finite element scheme for the active fluid model. The scheme employs an auxiliary variable to reformulate the fourth-order derivative term, an implicit-explicit treatment to deal with the nonlinear terms and the second-order pressure-projection method to split the velocity and pressure. Through rigorous theoretical analysis, the unique solvability, unconditional stability and error estimates of the numerical scheme are obtained. Then, several numerical experiments are presented to verify the efficiency and accuracy of the proposed scheme. Finally, the comparison of simulation results with laboratory results, including the motion direction of active fluid changes from disorder to order and reversal in 2D and 3D, demonstrate that the scheme can accurately capture and handle the complex dynamics of active fluid motion.