Numerical solution of the stationary Navier-Stokes equations using a multilevel finite element method

The numerical implementation of a multilevel finite element method for the steady-state Navier-Stokes equations is considered. The multilevel method proposed here for the Navier-Stokes equations is a multiscale method in which the full nonlinear Navier-Stokes equations are only solved on a single coarse grid; subsequent approximations are generated on a succession of refined grids by solving a linearized Navier-Stokes problem. Two numerical examples are considered: the first is an example for which an exact solution is known and the second is the driven cavity problem. We demonstrate numerically that for an appropriate choice of grids, a two- or three-level finite element method is significantly more efficient than the standard one-level finite element method.