A stabilized difference finite element method for the 3D steady Stokes equations

A stabilized difference finite element (SDFE) method based on the finite element pair ((P-1, P-1, P-1) x P-1) x (P-1 x P-0) is presented for the 3D steady Stokes equations. The difference finite element method consists of combining the finite difference discretization based on the P-1 x P-0-element in the z-direction and the finite element discretization based on the (P-1, P-1, P-1) x P-1-element in the (x, y)-plane. In this way, the numerical solution of the 3D steady Stokes equations can be transmitted into a series of the finite element solution pair (w(h)(k) , p(h)(k)) of the 2D steady Stokes equations and the finite element solution u(3h)(k) of the elliptic equation. The core of the stabilized method is to characterize the Ladyzhenskaya-Babuska-Brezzi "deficiency" of the unstable finite element pair with an appropriate operator, and the application of the operator in the stabilized mixed variational equations. The rigorous stability analysis and error estimation are developed, showing that the SDFE method is stable and optimally convergent. Several numerical tests are presented, confirming the theoretical predictions and verifying the accuracy of the considered method. (C) 2022 The Author(s). Published by Elsevier Inc.