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

Abstract

AbstractThis 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.