A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods

Abstract

A Galerkin finite element method is considered to approximate the incompressible Navier–Stokes equations together with iterative methods to solve a resulting system of algebraic equations. This system couples velocity and pressure unknowns, thus requiring a special technique for handling. We consider the Navier–Stokes equations in velocity––kinematic pressure variables as well as in velocity––Bernoulli pressure variables. The latter leads to the rotation form of nonlinear terms. This form of the equations plays an important role in our studies. A consistent stabilization method is considered from a new view point. Theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied.