This work analyzes a least-squares method in order to solve implicit time schemes associated to the 2D and 3D Navier–Stokes system, introduced in 1979 by Bristeau, Glowinksi, Periaux, Perrier and Pironneau. Implicit time schemes reduce the numerical resolution of the Navier–Stokes system to multiple resolutions of steady Navier–Stokes equations. We first construct a minimizing sequence (by a gradient type method) for the least-squares functional which converges strongly and quadratically toward a solution of a steady Navier–Stokes equation from any initial guess. The method turns out to be related to the globally convergent damped Newton approach applied to the Navier–Stokes operator. Then, we apply iteratively the analysis on the fully implicit Euler scheme and show the convergence of the method uniformly with respect to the time discretization. Numerical experiments for 2D examples support our analysis.

这项工作分析了最小二乘法，以解决与Bristeau，Glowinksi，Periaux，Perrier和Pironneau于1979年引入的2D和3D Navier-Stokes系统相关的隐式时间方案。隐式时间方案将Navier–Stokes系统的数值分辨率降低为稳态Navier–Stokes方程的多种分辨率。我们首先为最小二乘函数构造一个最小化序列（通过梯度类型方法），该函数可以从任何初始猜测中强烈求平方地求出稳定的Navier-Stokes方程的解。事实证明，该方法与应用于Navier-Stokes算子的全局收敛阻尼牛顿法有关。然后，我们对完全隐式Euler方案进行迭代分析，并针对时间离散化统一显示了该方法的收敛性。