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

TL;DR

This paper presents a novel space-time least-squares method for solving the unsteady Navier-Stokes equations, demonstrating strong convergence and quadratic rates in 2D and 3D cases, supported by numerical experiments.

Contribution

It introduces a globally convergent least-squares approach for the Navier-Stokes system with proven convergence properties and connections to damped Newton methods.

Findings

Strong convergence of the method for 2D and 3D cases.

Quadratic convergence after finite iterations related to viscosity.

Numerical experiments validate the theoretical analysis.

Abstract

We introduce and analyze a space-time least-squares method associated to the unsteady Navier-Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess, we construct a minimizing sequence for the least-squares functional which converges strongly to a solution of the Navier-Stokes system. After a finite number of iterates related to the value of the viscosity constant, the convergence is quadratic. Numerical experiments within the two dimensional case support our analysis. This globally convergent least-squares approach is related to the damped Newton method when used to solve the Navier-Stokes system through a variational formulation.