DIFFERENCE FINITE ELEMENT METHOD FOR THE 3D STEADY NAVIER-STOKES EQUATIONS

In this work, a difference finite element method for the 3D steady Navier-Stokes equations is presented. This new method consists of transmitting the finite element solution (uh,ph) of the three-dimensional (3D) steady Navier--Stokes equations into a series of the finite element solutions (unk h ,pnk h ) of the 2D steady Oseen iterative equations, which are solved by using the finite element pair (P1b, P1b, P1) \times P1 satisfying the discrete inf-sup condition in a 2D domain \omega . In addition, we use finite element pair ((P1 b ,P1 b ,P1) \times P1) \times (P1 \times P0) to solve the 3D steady Oseen iterative equations, where the velocity-pressure pair satisfies the discrete inf-sup condition in a 3D domain \Omega under the quasi-uniform mesh condition. To overcome the difficulty of nonlinearity, we apply the Oseen iterative method and present the weak formulation of the difference finite element method for solving the 3D steady \tau Oseen iterative equations. Moreover, we provide the existence and uniqueness of the difference finite element solutions (unh,pnh) = (\sumkl3 =0 unk h \phik(z),\sumkl3=1pnkh \psik(z)) of the 3D steady Oseen iterative equations and deduce the first order convergence with respect to (\sigman+1,\tau ,h) of the difference finite element solutions (unh, pnh) to the exact solution (u, p) of the 3D steady Navier-Stokes equations. Finally, some numerical tests are presented to show the accuracy and effectiveness of the proposed method.