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

Jérôme Lemoine (1) , Arnaud Munch (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

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, in contrast to standard Newton method used to solve the weak formulation of the Navier-Stokes system. 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.

Keywords: Least-squares approach; Steady Navier-Stokes system; Gradient method; Steady Navier-Stokes equation; Implicit time scheme; Damped Newton method AMS subject classifications 35Q30; 93E24; Leastsquares approach; Damped Newton method

Domains: Optimization and Control [math.OC]

Jérôme Lemoine, Arnaud Munch. Resolution of the implicit Euler scheme for the Navier-Stokes equation through a least-squares method. Numerische Mathematik, 2021, 147 (2), pp.349-391. ⟨hal-01996429v4⟩