An L² finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.