Difference Finite Element Methods Based on Different Discretization Elements for the Four-Dimensional Poisson Equation

This paper proposes difference finite element (DFE) methods for the Poisson equation in a four-dimensional (4D) region $ω × (0, L_4 ).$ The method converts the Poisson equation in a 4D region into a series of three-dimensional (3D) subproblems by the finite difference discretization in $(0, L_4)$ and deals with the 3D subproblems by the finite element discretization in $ω.$ In performing the finite element discretization, we select different discretization elements in the region $ω:$ hexahedral, pentahedral, and tetrahedral elements. Moreover, we prove the stability of the DFE solution $u_h$ and deduce the first-order convergence of $u_h$ with respect to the exact solution $u$ under $H^1$-error. Finally, three numerical examples are given to verify the accuracy and effectiveness of the DFE method.