Difference finite element method for the 3D steady Stokes equations

In this paper, a difference finite element (DFE) method is presented for the 3D steady Stokes equations. This new method consists of transmitting the finite element solution ( u h , p h ) of the 3D steady Stokes equations in the direction of ( x , y , z ) into a series of the finite element solution ( u h k , p h k ) of the 2D steady Stokes equations. Here the 2D steady Stokes equations are solved by the finite element space pair ( P 1 b , P 1 b , P 1 ) × P 1 , where the 2D finite element pair ( P 1 b , P 1 b ) × P 1 satisfies the discrete inf-sup condition in a 2D domain ω . Here we design the weak formulation of the DFE method based on the 3D finite element pair ( ( P 1 b , P 1 b , P 1 ) × P 1 ) × ( P 1 × P 0 ) under the quasi-uniform mesh condition, prove that the 3D finite element pair satisfies the discrete inf-sup condition in a 3D domain Ω and provide the existence, uniqueness and stability of the DFE solution ( u h , p h ) = ( ∑ k = 0 l 3 u h k ϕ k ( z ) , ∑ k = 1 l 3 p h k ψ k ( z ) ) and deduce the first order convergence of the DFE solution ( u h , p h ) with respect to the exact solution ( u , p ) of the 3D steady Stokes equations. Finally, some numerical tests are presented to show the accuracy and efficiency for the proposed method.