Analysis of the linearly extrapolated BDF2 fully discrete Modular Grad-div stabilization method for the micropolar Navier-Stokes equations

We investigate a fully discrete modular grad-div (MGD) stabilization algorithm for solving the incompressible micropolar Navier-Stokes equations (IMNSE) model, which couples the incompressible Navier-Stokes equations and the angular momentum equation together. The mixed finite element (FE) method is applied for the spatial discretization. The time discretization is based on the BDF2 implicit scheme for the linear terms and the two-step linearly extrapolated scheme for the convective terms. The considered algorithm constitutes two steps, which involve a post-processing step for linear velocity. First, we decouple the fully coupled IMNSE model into two smaller sub-physics problems at each time step (one is for the linear velocity and pressure, the other is for the angular velocity), which reduces the size of the linear systems to be solved and allows for parallel computing of the two sub-physics problems. Then, in the post-processing step, we only need to solve a symmetrical positive determined grad-div system of linear velocity at each time step, which does not increase the computational complexity by much. However, the post-processing step can improve the solution quality of linear velocity. Moreover, we obtain unconditional stability, and error estimates of the linear velocity and angular velocity. Finally, several numerical experiments involving three-dimensional and two-dimensional settings are used to validate the theoretical findings and demonstrate the benefits of the modular grad-div (MGD) stabilization algorithm.