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 omega x (0, L4). 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, L4) and deals with the 3D subproblems by the finite element discretization in omega. In performing the finite element discretization, we select different discretization elements in the region omega: hexahedral, pentahedral, and tetrahedral elements. Moreover, we prove the stability of the DFE solution uh and deduce the first-order convergence of uh with respect to the exact solution u under H1-error. Finally, three numerical examples are given to verify the accuracy and effectiveness of the DFE method.