几乎不可压缩线性弹性问题的多重网格Uzawa型混合有限元方法

数学物理学报 ›› 2018, Vol. 38 ›› Issue (5): 873-882.

几乎不可压缩线性弹性问题的多重网格Uzawa型混合有限元方法

葛志昊^1,^2,*(),葛媛媛¹()

¹ 河南大学数学与统计学院河南开封 475004
² 河南大学应用数学研究所河南开封 475004

收稿日期:2017-07-11 出版日期:2018-11-09 发布日期:2018-11-09
通讯作者: 葛志昊 E-mail:zhihaoge@henu.edu.cn;geyuan920@126.com
作者简介:葛媛媛, E-mail:geyuan920@126.com
基金资助:
河南省自然科学基金(162300410031);河南大学优秀青年培育项目(yqpy20140039)

Multigrid Uzawa-Type Mixed Finite Element Methods for Nearly Incompressible Linear Elasticity Problem

Zhihao Ge^1,^2,*(),Yuanyuan Ge¹()

¹ School of Mathematics and Statistics, Henan University, Henan Kaifeng 475004
² Institute of Applied Mathematics, Henan University, Henan Kaifeng 475004

Received:2017-07-11 Online:2018-11-09 Published:2018-11-09
Contact: Zhihao Ge E-mail:zhihaoge@henu.edu.cn;geyuan920@126.com
Supported by:
the Natural Science Foundation of Henan Province(162300410031);the Excellent Youth Program of Henan University(yqpy20140039)

摘要/Abstract

摘要：

该文针对几乎不可压缩弹性问题，设计了多重网格Uzawa型混合有限元方法，成功克服了"闭锁"现象.通过引入"压力"变量p将弹性问题转化为一个鞍点型系统，对该系统将Uzawa型迭代法和多重网格方法相结合，建立了多重网格和套迭代多重网格Uzawa型混合有限元方法，并给出了该算法的收敛性.数值算例验证了方法的有效性和稳定性.

关键词: 几乎不可压缩弹性问题, Uzawa型混合有限元方法, 多重网格方法

Abstract:

In this paper, we propose two new multigrid Uzawa-type mixed finite element methods for the nearly incompressible elasticity problem, which could overcome the 'locking' phenomenon. By introducing an extra pressure variable, we reformulate the elasticity problem into a saddle-point system, and by coupling the Uzawa-type method with multigrid methods, we develop two effective iteration methods:multigrid Uzawa-type mixed finite element method and nested iteration multigrid Uzawa-type mixed finite element method. Also, we present the convergent results of the algorithms. The methods are locking-free and stable for any finite element pairs spaces. Finally, we give some numerical examples to verify the theoretical results of the paper.

Key words: Nearly incompressible elasticity problems, Uzawa-type mixed finite element method, Multigrid methods

中图分类号:

O241.82

葛志昊,葛媛媛. 几乎不可压缩线性弹性问题的多重网格Uzawa型混合有限元方法[J]. 数学物理学报, 2018, 38(5): 873-882.

Zhihao Ge,Yuanyuan Ge. Multigrid Uzawa-Type Mixed Finite Element Methods for Nearly Incompressible Linear Elasticity Problem[J]. Acta mathematica scientia,Series A, 2018, 38(5): 873-882.

图/表 6

表 4.1

表 4.2

表 4.3

表 4.4

表 4.5

表 4.6

参考文献 8

1	Johnson C , Mercier B . Some equilibrium finite element methods for two dimensional elasticity problems. Numer Math, 1978, 30 (1): 103- 116 doi: 10.1007/BF01403910
2	Stenberg R . A family of mixed finite elements for the elasticity problem. Numer Math, 1988, 53 (5): 513- 538 doi: 10.1007/BF01397550
3	Aronld D N , Winther R . Mixed finite elements for elasticity. Numer Math, 2002, 92: 401- 419 doi: 10.1007/s002110100348
4	Auricchio F , Lovadina C . An analysis of some mixed-enhanced finite element for plane elasticity. Comput Methods Appl Mech Engrg, 2005, 194: 2947- 2968 doi: 10.1016/j.cma.2004.07.028
5	Falk R S . Finite element methods for linear elasticity. Lecture Notes in Mathematics, 2008, 1939 (2): 159- 194
6	Babuška I , Suri M . Locking effects in the finite element approximation of elasticity problems. Numer Math, 1992, 62 (1): 439- 463 doi: 10.1007/BF01396238
7	Babuška I , Suri M . On locking and robustness in the finite element method. SIAM J Numer Anal, 1992, 29 (5): 1261- 1293 doi: 10.1137/0729075
8	葛志昊,葛媛媛.几乎不可压弹性问题的Uzawa型自适应有限元方法,计算数学(已接收), 2017
	Ge Z H, Ge Y Y. New Uzawa-type adaptive finite element method for nearly incompressible elasticity problem (in Chinese). Accepted by Mathematica Numerica Sinica, 2017

$ \lambda $	$ \\|\vec{u}_h\\| $	$ \|\vec{u}-\vec{u}_h\|_{H^1} $	$ \\|\vec{u}-\vec{u}_h\\| $	$ \|\vec{u}-\vec{u}_h\| $	$ \frac{\\|\vec{u}-\vec{u}_h\\|}{\\|\vec{u}\\|} $
$ 10^{-1} $	0.00752572	0.00990784	0.000304453	0.0111987	0.0391521
$ 10^{0} $	0.00746715	0.00996155	0.000364112	0.0112154	0.0468241
$ 10^{3} $	0.00110679	0.0496878	0.00672073	0.0496914	0.864273
$ 10^{5} $	$ 1.3701\times{10^{-5}} $	0.0570505	0.00776319	0.0570505	0.998333
$ 10^{10} $	$ 1.37356\times{10^{-10}} $	0.0571429	0.00777616	0.0571429	1.000000

$ h $	$ \|\vec{u}-\vec{u}_h\| $	$\|\vec{u}-\vec{u}_h\|_{H^1}$	$\\|\vec{u}-\vec{u}_h\\|$	$ \\|p-p_h\\| $
$\sqrt{2}/2^4$	0.0112127	0.00990952	0.000289012	0.00148229
$\sqrt{2}/2^5$	0.00566004	0.00497397	7.3669$\times10^{-5}$	0.00979325
$\sqrt{2}/2^6$	0.00283683	0.0024893	1.85071$\times{10^{-5}}$	0.194218
$\sqrt{2}/2^7$	0.00141927	0.00124493	4.6324$\times10^{-6}$	0.000975646

$ h $	$ \|\vec{u}-\vec{u}_j\|_{H^1} $	$\|\vec{u}-\vec{u}_j\|$	$ \\|\vec{u}-\vec{u}_j\\| $	$ \\|p-p_j\\| $
$ \frac{\sqrt{2}}{2^4} $	0.00990824	0.0112103	0.000289378	0.000981775
$ \frac{\sqrt{2}}{2^5} $	0.00497386	0.00565981	7.37462$\times10^{-5}$	0.000255086
$ \frac{\sqrt{2}}{2^6} $	0.00248929	0.00283681	1.85207$\times10^{-5}$	6.36543$\times10^{-5}$
$ \frac{\sqrt{2}}{2^7} $	0.00124493	0.00141927	4.63482$\times10^{-6}$	1.57971$\times10^{-5}$
$ \frac{\sqrt{2}}{2^8} $	0.000622503	0.00070974	1.15892$\times10^{-6}$	3.9281$\times10^{-6}$
平均误差阶	0.9981	0.9954	1.9910	1.9914

迭代次数	$ \|\vec{u}-\vec{u}_h\|_{H^1} $	$ \\|\vec{u}-\vec{u}_h\\| $	$ \|\vec{u}-\vec{u}_h\| $	$\\|p-p_h\\| $
10	0.9918	1.9162	0.9853	-
30	0.9945	1.9666	0.9892	-
100	0.9959	1.9819	0.9914	-
500	0.9969	1.9866	0.9930	-

	$ h $	$ \|\vec{u}-\vec{u}_j\| $	$\\|\vec{u}-\vec{u}_j\\|$	$\|\vec{u}-\vec{u}_j\|_{H^1}$	$ \\|p-p_j\\| $
MUMFEM	$\sqrt{2}/2^4$	0.011224	0.000289379	0.00991621	0.00858243
	$\sqrt{2}/2^5$	0.00567571	7.39909$\times10^{-5}$	0.00498314	0.00919007
	$\sqrt{2}/2^6$	0.00285473	1.88377$\times10^{-5}$	0.00249968	0.00957303
	$\sqrt{2}/2^7$	0.00143837	4.97168$\times10^{-6}$	0.00125595	0.00978083
平均收敛阶		0.9880	1.9543	0.9937
IMUMFEM	$\sqrt{2}/2^4$	0.0112356	0.00028971	0.00992303	0.00609284
	$\sqrt{2}/2^5$	0.00568659	7.40819$\times10^{-5}$	0.00498939	0.0073056
	$\sqrt{2}/2^6$	0.00286445	1.90212$\times10^{-5}$	0.00250526	0.00843153
	$\sqrt{2}/2^7$	0.00144767	5.25868$\times10^{-6}$	0.00126132	0.00915367
平均收敛阶		0.9854	1.9279	0.9919