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 \((P_1^b, P_1^b, P_1) \times P_1\) $(P_1^b,P_1^b,P_1)\times P_1$ in the direction of (x, y), where the finite element space pair \((P_1^b, P_1^b) \times P_1\) $(P_1^b,P_1^b)\times P_1$ satisfies the discrete inf-sup condition in a 2D domain \(\omega \) $\omega$ . This new method consists of transmitting the finite element solution \((u_h,p_h)\) $(u_h,p_h)$ of the 3D steady Navier-Stokes equations in the direction of (x, y, z) into a series of the finite element solution pair \((u_h^{nk},p_h^{nk})\) $(u_h^{\mathit{nk}},p_h^{\mathit{nk}})$ based on the 2D finite element space pair \((P_1^b, P_1^b, P_1)\times P_1\) $(P_1^b,P_1^b,P_1)\times P_1$ , 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 \(H^1-L^2\) $H^1-L^2$ -optimal error estimate with respect to \((\varepsilon , \sigma ^{n+1}, h, \tau )\) $(\epsilon ,{\sigma }^{n+1},h,\tau )$ of the numerical solution \((u^n_h,p_h^n)\) $(u_h^n,p_h^n)$ to the exact solution \((\tilde{u},\tilde{p})\) $(\tilde{u},\tilde{p})$ is provided. Here \(0<\varepsilon<<1\) $0<\epsilon <<1$ is a penalty parameter, \(\sigma =\frac{N}{\nu ^2}\Vert F\Vert _{-1,\Omega }\) $\sigma =\frac{N}{{\nu }^2}{\Vert F\Vert }_{-1,\mathrm{\Omega }}$ is the uniqueness index, n is a iterative step number, \(\tau \) $\tau$ 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.