The paper is devoted to the study and analysis of the mixed stiffness finite element method for the Navier-Stokes equations, based on a formulation of velocity-pressure-stress deviatorics. The method used low order Lagrange elements and leads to optimal error order of convergence for velocity, pressure, and stress deviatorics by means of mesh-dependent norms defined in this paper. The main advantage of the MSFEM is that the stream function can not only be employed to satisfy the divergence constraint but stress deviatorics can also be eliminated at the element level so that it is unnecessary to solve a larger algebraic system containing stress multipliers, or to develop a special code for computing the MSFE solutions of Navier-Stokes equations because we can use the computing codes used in solving the Navier-Stokes equations with the velocity-pressure formulation, or even the computing codes used in solving the problems of solid mechanics.