• 论文 •

### 几乎不可压线弹性问题的新的Uzawa型自适应有限元方法

1. 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）.

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

1. 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

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.

MR(2010)主题分类:

()
 [1] Johnson C, Mercier B. Some equilibrium finite element methods for two dimensional elasticity problems[J]. Numer. Math., 1978, 30(1):103-116.[2] Stenberg R. A family of mixed finite elements for the elasticity problem[J]. Numer. Math., 1988, 53(5):513-538.[3] Falk R. Noncomforming finite element methods for the equations of linear elasticity[J]. Math. Comp., 1991, 57:529-550.[4] Aronld D, Winther R. Mixed finite elements for elasticity[J]. Numer. Math., 2002, 92:401-419.[5] Auricchio F, Lovadina C. An analysis of some mixed-enhanced finite element for plane elasticity[J]. Comput. Meth. Appl. Mech. Engrg, 2005, 194:2947-2968.[6] Falk R. Finite Element Methods for Linear Elasticity[M]. Lecture Notes in Mathematics, Springerverlag, 2008.[7] Zhang Z. Analysis of some quadrilateral nonconforming elements for incompressible elasticity[J]. SIAM J. Numer. Anal., 1997, 34(2):640-663.[8] Brink U, Stein E. On some mixed finite element methods for incompressible and nearly incompressible finite elasticity[J]. Comput. Mech., 1996, 19(1):105-119.[9] Lamichhane B. Inf-sup stable finite element pairs based on dual meshes and bases for nearly incompressible elasticity[J]. IMA J. Numer. Anal., 2009, 29(2):404-420.[10] Lamichhane B. A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids[J]. J. Comput. Appl. Math., 2014, 260(2):356-363.[11] Babuška I, Suri M. Locking effects in the finite element approximation of elasticity problems[J]. Numer. Math., 1992, 62(1):439-463.[12] Babuška I, Suri M. On locking and robustness in the finite element method[J]. Numer. Anal., 1990, 29(5):1261-1293.[13] Herrmann L. Elasticity equations for incompressible and nearly incompressible materials by a variational theorem[J]. AIAA J., 2015, 3(10):1896-1900.[14] Bänsch E, Morin P. An adaptive Uzawa FEM for the stokes problem:convergence without the inf-sup condition[J]. Numer. Anal., 2002, 40:1207-1229.
 [1] 唐斯琴, 李宏, 董自明, 赵智慧. 对流反应扩散方程的稳定化时间间断时空有限元解的误差估计[J]. 计算数学, 2020, 42(4): 472-486. [2] 洪庆国, 刘春梅, 许进超. 一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用[J]. 计算数学, 2020, 42(3): 298-309. [3] 关宏波, 洪亚鹏. 抛物型界面问题的变网格有限元方法[J]. 计算数学, 2020, 42(2): 196-206. [4] 李世顺, 祁粉粉, 邵新平. 求解定常不可压Stokes方程的两层罚函数方法[J]. 计算数学, 2019, 41(3): 259-265. [5] 王芬玲, 樊明智, 赵艳敏, 史争光, 石东洋. 多项时间分数阶扩散方程各向异性线性三角元的高精度分析[J]. 计算数学, 2018, 40(3): 299-312. [6] 张纯禹, 陈恭, 王一正, 王烨. 快速求解参数化偏微分方程的缩减基有限元方法及其在核工程中的应用[J]. 计算数学, 2017, 39(4): 431-444. [7] 司红颖, 魏先勇, 陈绍春. 二阶椭圆问题的一类广义有限元法[J]. 计算数学, 2016, 38(4): 405-411. [8] 胡俊, 石钟慈. Reissner-Mindlin板问题带约束非协调旋转Q1有限元方法[J]. 计算数学, 2016, 38(3): 325-340. [9] 宋海明, 张琪, 李景治, 刘宏宇. 求解美式回望期权的有限元方法[J]. 计算数学, 2016, 38(3): 245-256. [10] 郑权, 高玥, 秦凤. Helmholtz方程外边值问题的基于修正的DtN边界条件的有限元方法[J]. 计算数学, 2016, 38(2): 200-211. [11] 刘会坡. 中子输运方程误差估计及自适应计算[J]. 计算数学, 2015, 37(3): 264-272. [12] 杨宇博, 祝鹏, 尹云辉. 分层网格上奇异摄动问题的一致NIPG分析[J]. 计算数学, 2014, 36(4): 437-448. [13] 司红颖, 陈绍春. 半线性椭圆问题Petrov-Galerkin逼近及亏量迭代[J]. 计算数学, 2014, 36(3): 316-324. [14] 周琴, 潘雪琴, 冯民富. 对流占优的Sobolev方程的投影稳定化有限元方法[J]. 计算数学, 2014, 36(1): 99-112. [15] 祝鹏, 尹云辉, 杨宇博. 奇异摄动问题内罚间断有限方法的最优阶一致收敛性分析[J]. 计算数学, 2013, 35(3): 323-336.