A Stabilized Difference Finite Element Method for the 3D Steady Incompressible Navier-Stokes Equations

In this paper, we develop a stabilized difference finite element (SDFE) method for the 3D steady incompressible Navier-Stokes equations and apply Oseen iterative method to deal with the nonlinear term. Firstly, the finite difference discretization based on finite element pair P-1 x P-0 in the z-direction is used to obtain the finite difference solution (u(tau)(n) = Sigma(l3)(k=0) u(nk )(x, y)phi(k)(z), p(tau)(n) = Sigma(l3)(k=1) p(nk)(x, y)psi(k)(z)), where (u(nk), p(nk)) is the solution of 2D linearized Navier-Stokes equations, and then the stabilized finite element discretization based on finite element pair (P-1, P-1, P-1) x P-1 in the (x, y)-plane is used to approximate (u(nk), p(nk)), so as to obtain the SDFE solution (u(h)(n) = Sigma(l3)(k=0) u(h)(nk )(x, y)phi(k)(z), p(h)(n) = Sigma(l3)(k=1) p(h)(nk)(x, y)psi(k)(z)) of the 3D linearized Navier-Stokes equations. Our method has the following features. First, difference finite element method overcomes the difficulty of the 3D space discretization. Second, the stabilized method does not require specification of mesh-dependent parameters and retain the symmetry of the original equations. The rigorous stability analysis and error estimate are developed, showing that SDFE method is stable and has optimal convergence. Several numerical tests are presented, confirming the theoretical predictions and verifying the accuracy of the considered method.