Numerical Solution of the Stationary Navier--Stokes Equations Using a Multilevel Finite Element Method

Abstract

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.