A new finite element method is presented for use of quadrilateral nine-node elements in the solution of the incompressible Navier–Stokes equations. In a conventional predictor–corrector scheme, the method applies the nodal averaging technique to discretize the Poisson equation used for the simultaneous relaxation of velocity and pressure. Additionally, efficient approximation procedures are devised to increase the speed of computation without deteriorating solution accuracy. The proposed numerical schemes are evaluated on two-dimensional test problems including a classical lid-driven cavity flow and a flow over a backward-facing step in a flow channel. The results show good accuracy even when distorted elements are used for calculation.