Finite-element model in vorticity and current function for the numerical solution of the Navier-Stokes equations

A numerical procedure for the integration of the incompressible Navier-Stokes equations, when expressed in terms of a stream function equation and a vorticity transport equation, is presented. This procedure comprises the variational formulation of the equations, the construction of the approximation spaces by the finite element method and the discretization via the Galerkin method. For the stationary problems, the system of non-linear algebraic equations resulting from the discretization is solved by the Newton-Raphson algorithm. Finally, for the transient problems, the solution of the non-linear ordinary differential equations resulting from the spatial discretization is accomplished through a Crank-Nicolson scheme.