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

Abstract

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

CLC Number:

O241.82

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.

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References 8

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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

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

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	$ h $	$ \|\vec{u}-\vec{u}_j\| $	$\\|\vec{u}-\vec{u}_j\\|$	$\|\vec{u}-\vec{u}_j\|_{H^1}$	$ \\|p-p_j\\| $
有校正过程	$1/2^3$	0.0178683	0.000743724	0.0160728	0.0121122
	$1/2^4$	0.00956379	0.000207486	0.00873815	0.00771615
	$1/2^5$	0.00439364	4.30938$\times10^{-5}$	0.00400727	0.00466195
	$1/2^6$	0.00216383	1.04478$\times10^{-5}$	0.0019715	0.00192831
	$1/2^7$	0.00107643	$ 2.55895\times10^{-6}$	0.00098056	0.000353896
无校正过程	$1/2^3$	0.0178683	0.000743724	0.0160728	0.0121122
	$1/2^4$	0.00956379	0.000207486	0.00873815	0.00771615
	$1/2^5$	0.00440023	4.33507$\times10^{-5}$	0.00401131	0.00632012
	$1/2^6$	0.00217538	1.07786$\times10^{-5}$	0.00197805	0.00593642
	$1/2^7$	0.00108893	$ 3.03289\times10^{-6}$	0.000987726	0.0057875