Convergence analysis of a new multiscale finite element method with the element for the incompressible flow

In this paper, we propose a new multiscale finite element method for the stationary Navier–Stokes problem. This new method for the lowest order finite element pairs P 1 / P 0 is based on the multiscale enrichment and derived from the Navier–Stokes problem itself. Therefore, the new multiscale finite element method better reflects the nature of the nonlinear problem. The well-posedness of this new discrete problem is proved under the standard assumption. Meanwhile, convergence of the optimal order in H 1 -norm for velocity and L 2 -norm for pressure is obtained. Especially, via applying a new dual problem for the incompressible Navier–Stokes problem and some techniques in the process for proof, we establish the convergence of the optimal order in L 2 -norm for the velocity. Finally, numerical examples confirm our theory analysis for this new multiscale finite element method and validate the high effectiveness of this new method.