Weak approximation of the ψ-ω equations with explicit viscous diffusion

This paper describes a variational formulation for solving the 2-D time-dependent incompressible Navier-Stokes equations expressed in the stream function and vorticity. The difference between the proposed approach and the standard one is that the vorticity equation is interpreted as an evolution equation for the stream function while the Poisson equation is used as an expression for evaluating the distribution of vorticity in the domain and on the boundary. A time discretization is adopted with the viscous diffusion made 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 that is based on a Taylor series expansion of the stream function near the boundary. Some conditional stability results and error estimates are given.