The IGR method enables us to study the singular structures of the blow up solutions of Prandtl equations. The numerical solutions to the incompressible Navier-Stokes equations with Navier boundary conditions are discussed. An unconditionally stable time discretization which is implicit in viscosity and explicit in both pressure and convection terms and finite difference discretization with local pressure boundary condition are employed. A two level preconditioned conjugate gradient method is introdeced to solve the elliptic type system.

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