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

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications Pub Date : 2024-08-23 DOI:10.1016/j.camwa.2024.08.016

Yaru Liu , Yinnian He , Dongwoo Sheen , Xinlong Feng

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引用次数: 0

Abstract

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.

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基于符合 P1(x,y)×Q1(z,s)元素的差分有限元法，用于四维泊松方程

本文提出了一种求解四维（4D）域 Ω 中泊松方程的差分有限元方法（DFEM）。该方法将第三和第四方向基于 Q1 元素的有限差分离散化与其他方向基于 P1 元素的有限元离散化相结合。通过这种方法，4D 泊松方程的数值解可以转化为一系列 2D 泊松方程的有限元解。此外，我们还证明了 DFE 解 uH 满足 H1 稳定性，误差函数 uH-u 在 H1 误差下实现了一阶收敛。最后，我们提供了三个数值示例来验证该方法的准确性和高效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Computers & Mathematics with Applications 工程技术-计算机：跨学科应用

CiteScore

5.10

自引率

10.30%

发文量

396

审稿时长

9.9 weeks

期刊介绍： Computers & Mathematics with Applications provides a medium of exchange for those engaged in fields contributing to building successful simulations for science and engineering using Partial Differential Equations (PDEs).