Resolution of the implicit Euler scheme for the Navier–Stokes equation through a least-squares method

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