P1$$ {P}_1 $$ –Nonconforming quadrilateral finite element space with periodic boundary conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Numerical Methods for Partial Differential Equations Pub Date : 2023-04-29 DOI:10.1002/num.23023

Jaeryun Yim, D. Sheen

引用次数: 1

Abstract

The P1$$ {P}_1 $$ –nonconforming quadrilateral finite element space with periodic boundary conditions is investigated. The dimension and basis for the space are characterized by using the concept of minimally essential discrete boundary conditions. We show that the situation is different based on the parity of the number of discretizations on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary conditions. Some of these numerical schemes are related to solving linear equations consisting of non‐invertible matrices. By courtesy of the Drazin inverse, the existence of corresponding numerical solutions is guaranteed. The theoretical relation between the numerical solutions is derived, and it is confirmed by numerical results. Finally, the extension to the three dimensions is provided.

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P1$${P}_1$$–具有周期边界条件的非协调四边形有限元空间：第一部分：尺寸、基、求解器和误差分析的基本结果

P1$${P}_1研究了具有周期边界条件的$$–非协调四边形有限元空间。空间的维度和基础是通过使用最小本质离散边界条件的概念来表征的。我们表明，基于坐标上离散化数量的奇偶性，情况是不同的。在对空间分析的基础上，我们提出了求解具有周期边界条件的椭圆问题的几种数值格式。其中一些数值格式与求解由不可逆矩阵组成的线性方程组有关。借助于Drazin逆，保证了相应数值解的存在性。推导了数值解之间的理论关系，并用数值结果加以证实。最后，提供了对三个维度的扩展。

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

Numerical Methods for Partial Differential Equations 数学-应用数学

CiteScore

7.20

自引率

2.60%

发文量

审稿时长

9 months

期刊介绍： An international journal that aims to cover research into the development and analysis of new methods for the numerical solution of partial differential equations, it is intended that it be readily readable by and directed to a broad spectrum of researchers into numerical methods for partial differential equations throughout science and engineering. The numerical methods and techniques themselves are emphasized rather than the specific applications. The Journal seeks to be interdisciplinary, while retaining the common thread of applied numerical analysis.