An efficient two-step algorithm for the incompressible flow problem

A new two-step stabilized finite element method for the 2D/3D stationary Navier–Stokes equations based on local Gauss integration is introduced and analyzed in this paper. The method consists of solving one Navier–Stokes problem based on the P1−P1 finite element pair and then solving a general Stokes problem based on the P2−P2 finite element pair, i.e., computes a lower order predictor and a higher order corrector. Moreover, the stability and convergence of the present method are deduced, which show that the new method provides an approximate solution with the convergence rate of the same order as the P2−P2 stabilized finite element solution solving the Navier–Stokes equations on the same mesh width. However, our method can save a large amount of computational time. Finally, numerical tests confirm the theoretical results of the method.