基于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

作　　者：袁驷[1] 和雪峰[1] 

机构地区：[1]清华大学土木工程系,北京100084

出　　处：《应用数学和力学》2006年第11期1280-1291,共12页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(50278046)

摘　　要：基于新近提出的一维有限元后处理超收敛算法———单元能量投影(EEP)法,将有限元自适应求解问题转化为对超收敛解答的自适应分段多项式插值问题;对于大多数问题,一步便可获得满意的有限元网格划分,在该网格上再次进行有限元计算,一般即可获得满足用户给定的误差限的有限元解答.即便未能完全满足精度要求,一般只需局部细分加密网格一至二步即可.该法简单实用、高效可靠,是一个颇具优势和潜力的自适应方法.以二阶椭圆型常微分方程模型问题为例,对该法的基本思想、实施策略及具体算法做一介绍,并给出有代表性的数值算例用以展示该法的优良性能和效果.Based on the newly-developed element energy projection （EEP） method for computation of super-convergent results in one-dimensional finite element method （FEM）, the task of serf-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy has been found to be very simple, rapid, cheap and efficient. Taking the el- liptical ordinary differential equation of the second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm were described. Representative numerical examples are 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 单元能量投影法 常微分方程 

分 类 号：O242.21[理学—计算数学]