A multi-level stabilized finite element method for the stationary Navier–Stokes equations

Abstract

This paper proposes and analyzes a multi-level stabilized finite element method for the two-dimensional stationary Navier–Stokes equations approximated by the lowest equal-order finite element pairs. The method combines the new stabilized finite element method with the multi-level discretization under the assumption of the uniqueness condition. The multi-level stabilized finite element method consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multi-level stabilized finite element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h j ∼ h j - 1 2 , j = 1 , … , J . Moreover, the numerical illustrations agree completely with the theoretical expectations. Therefore, the multi-level stabilized finite element method is more efficient than the standard one-level stabilized finite element method.