Stabilized multiscale finite element method for the stationary Navier–Stokes equations

In the paper, a stabilized multiscale finite element method for the stationary incompressible Navier–Stokes equations is considered. The method is a Petrov–Galerkin approach based on the multiscale enrichment of the standard polynomial space enriched with the unusual bubble functions which no longer vanish on every element boundary for the velocity space. The stability of the P 1 – P 0 triangular element (or the Q 1 – P 0 quadrilateral element) is established. And the optimal error estimates of the stabilized multiscale finite element method for the stationary Navier–Stokes equations are obtained.