Weak approximation of the psi-omega equations with explicit viscous diffusion

This paper describes a variational formulation for solving the 2-D time-dep endent incompressible Navier-Stokes equations expressed in the stream funct ion and vorticity. The difference between the proposed approach and the sta ndard one is that the vorticity equation is interpreted as an evolution equ ation for the stream function while the Poisson equation is used as an expr ession for evaluating the distribution of vorticity in the domain and on th e boundary. A time discretization is adopted with the viscous diffusion mad e explicit, which leads to split the incompressibility from the viscosity. In some sense, the present method generalizes to the variational framework a well-known idea which is used in finite differences approximations and th at is based on a Taylor series expansion of the stream function near the bo undary. Some conditional stability results and error estimates are given.