A two-grid method with backtracking technique for the stream function form of the Navier-Stokes equations

A two-grid method based on modified Newton's iteration with backtracking technique is proposed for the stream function formulation of the stationary Navier-Stokes equations. The convergence rate and error estimate of the algorithm are established. It's proven that the asymptotically optimal approximation can be achieved by solving two small systems on the coarse mesh and two similar linear problems (same stiffness matrix only with different right-hand side) on the fine mesh. Compared with the fine space, the coarse space can be extremely coarse both in theory and in numerical applications.