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.