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In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier–Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh satisfies the discrete inf-sup condition in a 3D domain Ω. 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 (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier–Stokes equations in the H1−L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity u of the 3D steady Navier–Stokes equations in the L2 norm.

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