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

• Published in: Numerische Mathematik Vol. 147; no. 2; pp. 349 - 391
• Main Authors: Lemoine, Jérôme, Münch, Arnaud
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2021; Springer Nature B.V; Springer Verlag
• Subjects: Computational fluid dynamics; Convergence; Fluid flow; Least squares method; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Numerical Analysis; Numerical and Computational Physics; Optimization and Control; Simulation; Theoretical; Two dimensional analysis; 65K10; 35Q30; 49M15; Damped Newton method; 93E24; Implicit time scheme; Leastsquares approach; Steady Navier-Stokes system; Gradient method; Damped Newton method AMS subject classifications 35Q30; Least-squares approach; Steady Navier-Stokes equation
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-021-01171-1

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.