Optimal error estimates of penalty difference finite element method for the 3D steady Navier-Stokes equations

In this paper, a penalty difference finite element (PDFE) method is presented for the 3D steady Navier-Stokes equations by using the finite element space pair (P-1(b), P-1(b), P-1) x P-1 in the direction of (x, y), where the finite element space pair (P-1(b), P-1(b)) x P-1 satisfies the discrete inf-sup condition in a 2D domain omega. This new method consists of transmitting the finite element solution (u(h), p(h)) of the 3D steady Navier-Stokes equations in the direction of (x, y, z) into a series of the finite element solution pair (u(h)(nk), p(h)(nk)) based on the 2D finite element space pair (P-1(b), P-1(b), P-1) x P-1, which can be solved by the 2D decoupled penalty Oseen iterative equations. Moreover, the PDFE method of the 3D steady Navier-Stokes equations is well designed and the H-1 - L-2-optimal error estimatewith respect to (epsilon, sigma(n+1), h, tau) of the numerical solution (u(h)(n), p(h)(n)) to the exact solution ((u) over tilde, (p) over tilde) is provided. Here 0 < epsilon << 1 is a penalty parameter, sigma = N/nu(2) parallel to F parallel to(-1,Omega) is the uniqueness index, n is a iterative step number, tau is a mesh size in the direction of z and h is a mesh size in the direction of (x, y). Finally, numerical tests are presented to show the effectiveness of the PDFE method for the steady Navier-Stokes equations.