A cost-effective approach to the solution of 2D Navier-Stokes equations for incompressible fluid flow problems is presented. The aim is to reach a good compromise between numerical properties and computational efficiency. In order to achieve the set goal, the nonlinear convective terms are approximated by means of characteristics and spatial approximations of equal order are performed by polynomials of degree two. In this way, the computational kernels are reduced to elliptic ones for which solution very efficient techniques are available. The time-advancing is afforded by a fractional step method combined with a stabilization technique suitably simplified, so that the inf-sup condition is easily overcome. The algebraic systems generated by the new technique are solved by an iterative solver (Bi-CGSTAB), preconditioned by means of a suitable Schwarz additive scalable preconditioner. The properties of the new method have been confirmed from the comparison among the results obtained by it, and those obtained from other methods in the solution of some well known test problems. The obtained results, both in terms of accuracy and computational efficiency, make realistic the possibility to extend the method to 3D problems and to develop a multidomain approach.