几乎不可压线弹性问题的新的Uzawa型自适应有限元方法

doi:10.12286/jssx.2018.3.287

计算数学 ›› 2018, Vol. 40 ›› Issue (3): 287-298.DOI: 10.12286/jssx.2018.3.287

几乎不可压线弹性问题的新的Uzawa型自适应有限元方法

葛志昊, 葛媛媛

河南大学数学与统计学院 & 应用数学所, 开封 475004

收稿日期:2017-06-09 出版日期:2018-09-15 发布日期:2018-08-08
基金资助:
河南省自然科学基金（No：162300410031），河南大学优秀青年资助项目（No：yqpy20140039）.

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葛志昊, 葛媛媛. 几乎不可压线弹性问题的新的Uzawa型自适应有限元方法[J]. 计算数学, 2018, 40(3): 287-298.

Ge Zhihao, Ge Yuanyuan. NEW UZAWA-TYPE ADAPTIVE FINITE ELEMENT METHODS FOR NEARLY INCOMPRESSIBLE LINEAR ELASTICITY PROBLEM[J]. Mathematica Numerica Sinica, 2018, 40(3): 287-298.

NEW UZAWA-TYPE ADAPTIVE FINITE ELEMENT METHODS FOR NEARLY INCOMPRESSIBLE LINEAR ELASTICITY PROBLEM

Ge Zhihao, Ge Yuanyuan

School of Mathematics and Statistics & Institute of Applied Mathematics, Henan University, Kaifeng 475004, China

Received:2017-06-09 Online:2018-09-15 Published:2018-08-08

本文针对几乎不可压线弹性问题设计新的Uzawa型自适应有限元方法，该方法可克服“闭锁”现象.通过引入“压力”变量将弹性问题转化为一个鞍点系统，对该系统将Uzawa型迭代法和自适应有限元方法相结合，建立了Uzawa型自适应有限元方法，并给出了该算法的收敛性.该算法采用低阶协调有限元逼近空间变量，选取的有限元空间对无需满足离散的BB条件.最后，数值算例验证了理论结果的正确性.

关键词： 几乎不可压线弹性问题, 离散BB条件, Uzawa型自适应有限元

In this paper, we propose two new Uzawa-type finite element methods for nearly incompressible linear elasticity problem, which could overcome the locking phenomenon. By introducing an extra "pressure" variable, we reformulate the original problem into a saddlepoint system, then we propose the new Uzawa-type adaptive finite element methods, and give the convergent results of the new methods. Our method is locking-free for any pair of the finite element spaces including the pair of finite element spaces which does not satisfy the discrete BB condition. Finally, we present some numerical examples to verify the theoretical results.

Key words: Nearly incompressible linear elasticity, BB condition, Uzawa iterative method, adaptive finite element method

MR(2010)主题分类:

65N30
65Z05

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摘要

几乎不可压线弹性问题的新的Uzawa型自适应有限元方法

NEW UZAWA-TYPE ADAPTIVE FINITE ELEMENT METHODS FOR NEARLY INCOMPRESSIBLE LINEAR ELASTICITY PROBLEM

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