二维边值问题有限元离散系统病态问题的病因抑制方法  

A Method of Inhibiting Pathogeny for Ill-Condition Problems of the Finite Element Discrete Systems of 2D Boundary Value Problems

作　　者：张衡 张宏 游文杰 郑汉垣 

机构地区：[1]无损检测技术福建省高等学校重点实验室(福建技术师范学院),福建 福清 [2]福建技术师范学院大数据与人工智能学院,福建 福清 [3]湖南三一工业职业技术学院工程机械学院,湖南 长沙 [4]福建技术师范学院电子与机械工程学院,福建 福清 [5]龙岩学院传播与设计学院,福建 龙岩

出　　处：《应用数学进展》2021年第12期4162-4171,共10页Advances in Applied Mathematics

摘　　要：讨论二维Poisson方程边值问题离散系统的病态问题,针对三角剖分和和四角剖分,基于Lagrange形函数形成有限元离散系统的病态问题,提出病因抑制方法;给出该系统的病态结构、病态因子、去病因子;利用病态因子估计离散系统的条件数;利用去病因子为最优预条件子,精准抑制病态发作,该预条件子的使用,几乎不增加求解的计算量,预处理后离散系统保持正定对称,条件数关于离散系统规模一致有界。The ill conditioned problem of the finite element discrete system based on triangular subdivision, tetragonal subdivision and Lagrange shape function for 2D Poisson equation boundary value problem is discussed. The method of inhibiting pathogeny was proposed. The ill condition structure, ill condition factor and ill elimination factor of the system were given;and the condition number of the equation was estimated by the ill conditioned factor. The ill elimination factor was used as the optimal preconditioner to precisely suppress the ill condition. The use of the preconditioner hardly increases the calculation of iteration. After the pretreatment, the equations keep positive definite symmetry and the condition number is uniformly bounded with respect to the scale of equations.

关 键 词：病因抑制方法 病态结构 病态因子 去病因子 最优预条件子 

分 类 号：O17[理学—数学]