Abstract: A unified analysis is presented for the stabilized methods including the pressure projection method and the pressure gradient local projection method of conforming and nonconforming low-order mixed finite elements for the stationary Navier-Stokes equations. The existence and uniqueness of the solution and the optimal error estimates are proved.

Key words: asymptotic convergence, quadratic integer convex programming, stochastic level value approximation, cross-entropy method, Navier-Stokes equation, Ladyzhenskaya-Babu?ska-Brezzi (LBB) condition, low-order finite element, pressure projection method, pressure gradient local projection method