Iterative schemes for the non-homogeneous Navier–Stokes equations based on the finite element approximation

In this paper, we consider the stability and convergence of three iterative schemes for the non-homogeneous steady Navier–Stokes equations. As a nonlinear problem, we will get a nonlinear discrete system if approximating the non-homogeneous Navier–Stokes equations. After proving the stability and error estimates of the finite element method for the non-homogeneous Navier–Stokes equations, three iterative schemes are investigated for solving the resulted nonlinear discrete system. The stability and convergence conditions for these iterative methods are also analyzed, respectively. Furthermore, new results for the stop criterion are proved. Finally, we show some numerical experiments to illustrate the theoretical prediction.