Finite Element Iterative Methods for the 3D Steady Navier–Stokes Equations †

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair [Formula: see text]. The method consists of transmitting the finite element solution [Formula: see text] of the 3D steady Navier–Stokes equations into the finite element solution pairs [Formula: see text] based on the finite element space pair [Formula: see text] of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair [Formula: see text] satisfies the discrete inf-sup condition in a 3D domain [Formula: see text]. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution [Formula: see text] of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to [Formula: see text] of the FE solution [Formula: see text] to the exact solution [Formula: see text] of the 3D steady Navier–Stokes equations in the [Formula: see text] norm. Finally, we also give the convergence order with respect to [Formula: see text] of the FE velocity [Formula: see text] to the exact velocity u of the 3D steady Navier–Stokes equations in the [Formula: see text] norm.