Least-squares approaches for the 2D Navier-Stokes system

International audienceWe analyze a least-squares approach in order to approximate weak solutions of the 2D-Navier Stokes system. In a first part, we consider the steady case and introduce a quadratic functional based on a weak norm of the state equation. We construct a minimizing sequence for the functional which converges strongly to a solution of the equation. After a finite number of iterates related to the value of the viscosity constant, the convergence is quadratic, from any initial guess. We then apply iteratively the analysis on the backward Euler scheme associated to the unsteady Navier-Stokes equation and prove the convergence of the iterative process uniformly with respect to the time discretization. In a second part, we reproduce the analysis for the unsteady case by introducing a space-time least-squares functional. 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. Numerical experiments illustrate our analysis