平面弹性问题自适应有限元方法的收敛性分析  

Convergence of an Adaptive Finite Element Method for 2D Elasticity Problems

作　　者：刘春梅[1] 钟柳强[2] 舒适[3] 肖映雄[4] 

机构地区：[1]湖南科技学院数学与计算科学系计算数学研究所,湖南永州425199 [2]华南师范大学数学科学学院,广州510631 [3]湘潭大学数学与计算科学学院,湖南湘潭411105 [4]湘潭大学土木工程与力学学院,湖南湘潭411105

出　　处：《应用数学和力学》2014年第9期969-978,共10页Applied Mathematics and Mechanics

基　　金：湖南省自然科学基金(14JJ3135);国家自然科学基金(11201159);全国博士学位论文作者专项资金(201212);广东省高等学校优秀青年教师培养计划(Yq2013054);广州市珠江科技新星项目(2013J2200063)~~

摘　　要：针对平面弹性问题,首先采用基于最新顶点二分法的网格加密方法,给出一种不需要标记振荡项和加密单元、不需要满足"内节点"性质的自适应有限元方法.其次,通过对各层网格上解函数和误差指示子的分析,利用相邻网格层上解函数的正交性、解函数和真解函数的能量误差的上界估计、相邻网格层上误差指示子的近似压缩性等结果,从理论上严格证明了该自适应有限元方法是收敛的.最后数值实验验证了该自适应有限元方法是收敛的和鲁棒的.For 2D linear elasticity problems, firstly, a standard adaptive finite element method （AFEM） was developed based on the newest vertex bisection grid refinement, which was marked only according to the error estimators without special treatment of the oscillation terms and intended conformance to the interior node properties. Secondly, through analysis of the numerical solution functions and error indicators at all the grid levels, the AFEM was strictly proved to be convergent by means of the orthogonality between the numerical solution functions at adjacent grid levels, the upper bound estimation of the energy errors between the true solution functions and the numerical solution functions, and the approximate compressibility of the error indicators between adjacent grid levels. Finally, several numerical experiments confirm that the presented AFEM is convergent and robust.

关 键 词：平面弹性问题 自适应有限元方法 收敛性 

分 类 号：O241.8[理学—计算数学] O242[理学—数学]