Abstract: This paper is the first in a series of papers in which we present a new finite element method for incompressible fluid dynamics——a dimensional reductionmethod.We show that it saves a lot of computing time and storages using this method when a simple basis of Yh is available.in this paper, the essential algorithm is presented, and a very simpte basis of V^h is given for solving the homogeneous Navier-Stokes equations in a simply connected region in R² using quadratic conforming triangular elements for the velocity field and piecewish constant triangular elements for the pressure1.