Abstract
In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair . The method consists of transmitting the finite element solution of the 3D steady Navier–Stokes equations into the finite element solution pairs based on the finite element space pair of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair satisfies the discrete inf-sup condition in a 3D domain . 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 of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to of the FE solution to the exact solution of the 3D steady Navier–Stokes equations in the norm. Finally, we also give the convergence order with respect to of the FE velocity to the exact velocity u of the 3D steady Navier–Stokes equations in the norm.
Highlights
The incompressible Navier–Stokes equations reflect the basic mechanical law of viscous fluid flow, which have important implications in fluid mechanics
Lots of works are devoted to this problem, and the finite element methods, finite volume methods and finite difference methods are the most successful methods
From Theorems 3–5, we find that the three iterative finite element (FE) methods are H 1 -uniform stable and first-order convergent with respect to (ν, 1 − σ ); the three iterative FE methods are L2 -uniform stable and second-order convergent with respect to ν; the Stokes iterative FE method is simpler than the Oseen iterative FE method; the Oseen iterative FE method is simpler than the Newton iterative
Summary
The incompressible Navier–Stokes equations reflect the basic mechanical law of viscous fluid flow, which have important implications in fluid mechanics.
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