A finite element variational multiscale method based on two-grid discretization for the steady incompressible Navier–Stokes equations

Abstract

By combining the best algorithmic features of two-grid discretization method and a recent variational multiscale method, a two-level finite element variational multiscale method based on two local Gauss integrations for convection dominated incompressible Navier–Stokes equations is proposed and analyzed. In this method, a fully nonlinear Navier–Stokes problem is first solved on a coarse grid, and then a linear problem is solved on a fine grid to correct the coarse grid solution, where the numerical forms of the Navier–Stokes equations both on coarse and fine grids are stabilized by a stabilization term defined by the difference between a consistent and an under-integrated matrix of the velocity gradient. Error bounds of the approximate solution are analyzed. Algorithmic parameter scalings of the method are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the efficiency and promise of the method.