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

Abstract

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 L 2-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 L 2-approximation to the stationary incompressible Navier-Stokes equations. The validity of L 2-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.

A new fluid-structure coupling model: Case of a pipe under internal pressure

“…We consider an incompressible isothermal Newtonian 31 flow (density ρ = constant, viscosity μ = constant), with a velocity field V → = ( v r , v θ , v z ) , which gives us the Navier-Stokes equation as follows:…”