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

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

Year:    2025

Author:    Yaru Liu, Yinnian He, Xinlong Feng

East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 2 : pp. 415–438

Abstract

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.2023-233.200224

East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 2 : pp. 415–438

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:    4D Poisson equation difference finite element method hexahedral element pentahedral element tetrahedral element.

Author Details

Yaru Liu

Yinnian He

Xinlong Feng