We are interested in the numerical modelling of the incompressible Navier-Stokes equations, in two space dimensions. The approach considered consists in developing semi-discrete finite element schemes for appropriate relaxation models, which formally arise as hyperbolic approximations of the Navier-Stokes equations. Stability properties of the relaxation finite element schemes are derived from estimating suitable modifications of the standard energy functional, that is suggested by the presence of relaxation terms. These techniques are also applied to prove error estimates and deduce the convergence of relaxation finite element schemes to the (smooth) solutions of the incompressible Navier-Stokes equations.