A new approach, iteration projection method, is proposed to solve the saddle point problem obtained after the full discretization of the unsteady Navier-Stokes equations. The proposed method iterates projections in each time step with a proper convection form. We prove that the projection iterations converge with a certain parameter regime. Optimal iteration convergence can be achieved with the modulation of parameter values. This new method has several significant improvements over the Uzawa method and the projection method both theoretically and practically. First, when the iterative projections are fully convergent in each time step, the numerical velocity is weakly divergence free (pointwise divergence free in divergence free finite element methods), and the stability and error estimate are rigorously proven. With proper parameters, this method converges much faster than the Uzawa method. Second, numerical simulations show that with rather relaxed stopping criteria which require only a few iterations each time step, the numerical solution preserves stability and accuracy for high Reynolds numbers, where the convectional projection method would fail. Furthermore, this method retains the efficiency of the traditional projection method by decoupling the velocity and pressure fields, which splits the saddle point system into small elliptic problems. Three dimensional simulations with Taylor-Hood P2/P1 finite elements are presented to demonstrate the performance and efficiency of this method. More importantly, this method is a generic approach and thus there are many potential improvements and extensions of the iterative projection method, including utilization of various convection forms, association with stabilization techniques for high Reynolds numbers, applications to other saddle point problems.

翻译：本文提出了一种新的方法——迭代投影法，用于求解非定常Navier-Stokes方程全离散化后得到的鞍点问题。该方法在每个时间步中以合适的对流形式进行迭代投影。我们证明了在特定参数范围内，投影迭代是收敛的。通过调整参数值，可实现最优迭代收敛。与Uzawa方法和投影方法相比，新方法在理论和实践上均有若干显著改进。首先，当每个时间步中的迭代投影完全收敛时，数值速度是弱散度自由的（在散度自由有限元方法中为点态散度自由），并且稳定性和误差估计得到了严格证明。在适当参数下，该方法的收敛速度远快于Uzawa方法。其次，数值模拟表明，采用相当宽松的停止准则（每个时间步仅需少量迭代），数值解在高雷诺数下仍能保持稳定性和精度，而传统投影方法在此情况下会失效。此外，该方法通过解耦速度场和压力场将鞍点系统分解为多个小型椭圆问题，从而保留了传统投影方法的效率。我们使用Taylor-Hood P2/P1有限元进行了三维模拟，展示了该方法的性能和效率。更重要的是，该方法是一种通用性方法，因此迭代投影法具有诸多潜在的改进和扩展方向，包括利用各种对流形式、结合高雷诺数稳定化技术、以及应用于其他鞍点问题。