Resolution of implicit time schemes for the Navier-Stokes system through a least-squares method.

Jérôme Lemoine (1, 2) , Arnaud Munch (1, 2)

1 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

Implicit time schemes reduce the numerical resolution of the Navier-Stokes system to multiple resolutions of steady Navier-Stokes equations. We analyze a least-squares method, introduced by Glowinski in 1979, to solve the steady Navier-Stokes equation. Precisely, we show that any minimizing sequences (constructed by gradient type methods) for a least-squares functional converges strongly toward solutions, assuming the initial guess in an explicit ball dependent of the time step and of the viscosity constant. The resulting method is faster and more robust than the Newton method used to solve the weak variational formulation for the Navier-Stokes system. Numerical experiments support our analysis.

Mots clés: Steady Navier-Stokes system; Least-squares approach; Gradient method

Domaines: Optimisation et contrôle [math.OC]

Jérôme Lemoine, Arnaud Munch. Resolution of implicit time schemes for the Navier-Stokes system through a least-squares method.. 2019. ⟨hal-01996429v2⟩