Error estimate for a simple two-level discretization of stream function form of the Navier-Stokes equations

In this paper, a simplified two-level finite element method with backtracking is proposed for the stream function formulation of the Navier-Stokes equations. This method requires the solution of two small systems on the coarse space and one symmetric, positive linear problem on the refined space. Error analysis for the case of Clough-Tocher or Bogner-Fox-Schmit elements is presented and the optimal asymptotic estimation is obtained in the H2- and H1-norms. Finally, numerical examples are given to verify the theoretical analysis.