A difference finite element method based on the conforming P1(x,y)×Q1(z,s) element for the 4D Poisson equation

This paper proposes a difference finite element method (DFEM) for solving the Poisson equation in the four-dimensional (4D) domain Ω. The method combines finite difference discretization based on the $Q_1$-element in the third and fourth directions with finite element discretization based on the $P_1$-element in the other directions. In this way, the numerical solution of the 4D Poisson equation can be transformed into a series of finite element solutions of the 2D Poisson equation. Moreover, we prove that the DFE solution $u_H$ satisfies $H^1$-stability, and the error function $u_H-u$ achieves first-order convergence under the $H^1$-error. Finally, we provide three numerical examples to verify the accuracy and efficiency of the method.