基于最佳超收敛阶EEP法的一维有限元自适应求解  

Self-Adaptive Strategy for One-Dimensional Finite Element Method Based on EEP Method With Optimal Super-Convergence Order

作　　者：袁驷[1] 邢沁妍[1] 王旭[1] 叶康生[1] 龙驭球(推荐) 

机构地区：[1]清华大学土木工程系结构工程与振动教育部重点实验室,北京100084 [2]不详

出　　处：《应用数学和力学》2008年第5期533-543,共11页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(50678093);长江学者和创新团队发展计划资助项目(IRT00736)

摘　　要：基于新近提出的具有最佳超收敛阶的单元能量投影(EEP)超收敛算法,提出用具有最佳超收敛阶的EEP超收敛解对有限元解进行误差估计,用均差法进行网格划分,用拟有限元解进行多次遍历而不反复求解有限元真解,形成一套新型的一维有限元自适应求解策略.该法理论上简明清晰,算法上高效可靠,对于大多数问题,一步自适应迭代便可给出按最大模度量逐点满足误差限的有限元解答.以二阶椭圆型常微分方程模型问题为例,介绍了该法的基本思想、实施策略及具体算法,并给出具有代表性的数值算例,以展示该法的优良性能和效果.Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved serf-adaptive strategyategy for one-dimensional finite element method （FEM） was proposed. In the strategy, a posteriori errors were estimated by comparing FEM solutions to EEP super-convergent solulions with optimal order of super-convergence, meshes were refined by using error-averaging method, and quasi-FEM solutions were used to replace true FEM solutions in the adaptive process. This sWategy has been found to be simple, clear, efficient and reliable. For most of the problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in maxnorm. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and computational algorithm were described and representative numerical examples were given to show the effectiveness and reliability of the proposed approach.

关 键 词：有限元法 自适应求解 超收敛 最佳收敛阶 单元能量投影 凝聚形函数 

分 类 号：O242.21[理学—计算数学]