Convergence analysis of a new multiscale finite element method for the stationary Navier–Stokes problem

In this paper, we propose a new multiscale finite element method for the stationary Navier–Stokes problem. This new method for the lowest equal order finite element pairs P1/P1P1/P1 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 the H1H1-norm for the velocity and the L2L2-norm for the pressure is obtained. Especially, via applying a new dual problem and some techniques in the process for proof, we establish the convergence of the optimal order in the L2L2-norm for the velocity. Finally, numerical examples confirm our theory analysis and validate the effectiveness of this new method.