A difference finite element method based on nonconforming finite element methods for 3D elliptic problems

In this paper, a class of 3D elliptic equations is solved by using the combination of the finite difference method in one direction and nonconforming finite element methods in the other two directions. A finite-difference (FD) discretization based on \(P_1\)-element in the z-direction and a finite-element (FE) discretization based on \(P_1^{NC}\)-nonconforming element in the (x, y)-plane are used to convert the 3D equation into a series of 2D ones. This paper analyzes the convergence of \(P_1^{NC}\)-nonconforming finite element methods in the 2D elliptic equation and the error estimation of the \({H^1}\)-norm of the DFE method. Finally, in this paper, the DFE method is tested on the 3D elliptic equation with the FD method based on the \(P_1\) element in the z-direction and the FE method based on the Crouzeix-Raviart element, the \(P_1\) linear element, the Park-Sheen element, and the \(Q_1\) bilinear element, respectively, in the (x, y)-plane.