| 三维泊松方程边值问题3次Lagrange有限元方程的病态结构和最优预条件子 |
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| 引用本文： | 张衡,郑汉垣.三维泊松方程边值问题3次Lagrange有限元方程的病态结构和最优预条件子[J].高等学校计算数学学报,2020(2):120-132. |
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| 作者姓名： | 张衡 郑汉垣 |
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| 作者单位： | 无损检测技术福建省高等学校重点实验室（福建师范大学福清分校）;福建师范大学福清分校电子与信息工程学院;龙岩学院传播与设计学院 |
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| 基金项目： | 福建省自然科学基金(2014J01006,2015J01587)。 |
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| 摘 要： | 1引言在科学计算和工程应用中,偏微分方程大规模数值求解问题通常转化为病态(高条件数)的大规模稀疏线性方程组的求解问题,其条件数(病态)经常随着问题规模的增加而增加1],成为影响求解效率和精度的瓶颈因素,因此,在求解之前,使用预处理技术来减少方程组的病态,成为提高求解效率和精度的必要措施.
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| 关 键 词： | 预条件子 边值问题 偏微分方程 三维泊松方程 预处理技术 求解效率 求解问题 瓶颈因素 |
THE ILL-CONDITION STRUCTURE AND OPTIMAL PRECONDITIONER OF THE FINITE ELEMENT EQUATION OF 3D POISSON EQUATION BOUNDARY VALUE PROBLEMS WITH CUBIC LAGRANGE SHAPE FUNCTION |
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| Zhang Heng,Zheng Hanyuan.THE ILL-CONDITION STRUCTURE AND OPTIMAL PRECONDITIONER OF THE FINITE ELEMENT EQUATION OF 3D POISSON EQUATION BOUNDARY VALUE PROBLEMS WITH CUBIC LAGRANGE SHAPE FUNCTION[J].Numerical Mathematics A Journal of Chinese Universities,2020(2):120-132. |
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| Authors: | Zhang Heng Zheng Hanyuan |
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| Institution: | (Key Laboratory of Nondestructive Testing(Puqing Branch of Fujian Normal University),Fujian Province University,Fuqing 350300;School of Electronic and Information Engineering,Fuqing Branch of Fujian Normal University,Puqing 350300;School of Communication and Design,Longyan University,Longyan 364012) |
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| Abstract: | While using finite element method to solve 3 D Poisson equation boundary value problems,the formed finite element equations based on cubic Lagrange shape function are ill-condition sparse linear systems.Based on the idea of structural analysis,the Ill-condition mechanism and preconditioning principle of the equation are discussed in this paper.The definitions of the Ill-condition structure,the Ill-condition factors,and the eliminate ill factor of the equations are given.We propose a eliminate ill factor according to the Ill-condition structure,and treat this eliminate ill factor as the preconditioner.In addition,quantitative analysis for the proposed preconditioner is given,and the analysis results show that the eliminate ill factor is the optimal preconditioner and the positive definite symmetry of the equations is kept and the condition number is close to a constant after pretreatment without causing more computing. |
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| Keywords: | Ill-condition mechanism Ill-condition structure Ill-condition factor Eliminate ill factor Preconditioning |
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