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A new approach, iteration projection method, is proposed to solve the saddlepoint problem obtained after the full discretization of the unsteadyNavier-Stokes equations. The proposed method iterates projections in each timestep with a proper convection form. We prove that the projection iterationsconverge with a certain parameter regime. Optimal iteration convergence can beachieved with the modulation of parameter values.  This new method has several significant improvements over the Uzawa methodand the projection method both theoretically and practically. First, when theiterative projections are fully convergent in each time step, the numericalvelocity is weakly divergence free (pointwise divergence free in divergencefree finite element methods), and the stability and error estimate arerigorously proven. With proper parameters, this method converges much fasterthan the Uzawa method. Second, numerical simulations show that with ratherrelaxed stopping criteria which require only a few iterations each time step,the numerical solution preserves stability and accuracy for high Reynoldsnumbers, where the convectional projection method would fail. Furthermore, thismethod retains the efficiency of the traditional projection method bydecoupling the velocity and pressure fields, which splits the saddle pointsystem into small elliptic problems. Three dimensional simulations withTaylor-Hood P2/P1 finite elements are presented to demonstrate the performanceand efficiency of this method.  More importantly, this method is a generic approach and thus there are manypotential improvements and extensions of the iterative projection method,including utilization of various convection forms, association withstabilization techniques for high Reynolds numbers, applications to othersaddle point problems.

Iterative projection method for unsteady Navier-Stokes equations with high Reynolds numbers