Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier–Stokes equations

This article considers two kinds of two-level stabilized finite element methods based on local Gauss integral technique for the two-dimensional stationary Navier–Stokes equations approximated by the lowest equal-order elements which do not satisfy the inf–sup condition. The two-level methods consist of solving a small non-linear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the Navier–Stokes equations on a fine mesh for a related choice of mesh widths. Therefore, the two-level methods are of practical importance in scientific computation. Finally, the performance of two kinds of two-level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments. The conclusion is that simple two-level stabilized method is a viable choice for the lowest equal-order approximations of the stationary Navier–Stokes problem.