Analysis of two decoupled time-stepping finite-element methods for incompressible fluids with microstructure

In this article, two full discrete schemes of the micropolar Navier–Stokes equations (MNSE) are proposed and analysed, namely a fully decoupled scheme and a decoupled penalty-projection scheme. The fully decoupled scheme decouples the MNSE system into two sub-problems, one is the velocity–pressure system, the other is the angular velocity system, but still keeps unconditional stability with respect to the time step size. The stability and the optimal rate of convergence of the scheme are proved. Moreover, the decoupled penalty-projection method is derived based on the Chorin/Temam projection methods for the velocity–pressure system equipped with grad-div stabilization with parameter γ, we also prove that this scheme is unconditionally stable. In particular, we show that the solution of the decoupled penalty-projection method converges to the associated full decoupled method solution as γ → ∞. These time-stepping discretization schemes can be implemented efficiently in practice. Numerical experiments are given to validate the convergence rate predictions and demonstrate the efficiency of the new methods.