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We introduce and analyze a space-time least-squares method associated to theunsteady Navier-Stokes system. Weak solution in the two dimensional case andregular solution in the three dimensional case are considered. From any initialguess, we construct a minimizing sequence for the least-squares functionalwhich converges strongly to a solution of the Navier-Stokes system. After afinite number of iterates related to the value of the viscosity constant, theconvergence is quadratic. Numerical experiments within the two dimensional casesupport our analysis. This globally convergent least-squares approach isrelated to the damped Newton method when used to solve the Navier-Stokes systemthrough a variational formulation.

A fully space-time least-squares method for the unsteady Navier-Stokes system