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
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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.
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.
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.
引文
[1] Johnson C, Mercier B. Some equilibrium finite element methods for two dimensional elasticity problems.Numer Math, 1978, 30(1):103-116
[2] Stenberg R. A family of mixed finite elements for the elasticity problem. Numer Math, 1988, 53(5):513-538
[3] Aronld D N, Winther R. Mixed finite elements for elasticity. Numer Math, 2002, 92:401-419
[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
[5] Falk R S. Finite element methods for linear elasticity. Lecture Notes in Mathematics, 2008, 1939(2):159-194
[6] Babuska I, Suri M. Locking effects in the finite element approximation of elasticity problems. Numer Math, 1992, 62(1):439-463
[7] Babuska I, Suri M. On locking and robustness in the finite element method. SIAM J Numer Anal, 1992,29(5):1261-1293
[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),2017Ge 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
[2] Stenberg R. A family of mixed finite elements for the elasticity problem. Numer Math, 1988, 53(5):513-538
[3] Aronld D N, Winther R. Mixed finite elements for elasticity. Numer Math, 2002, 92:401-419
[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
[5] Falk R S. Finite element methods for linear elasticity. Lecture Notes in Mathematics, 2008, 1939(2):159-194
[6] Babuska I, Suri M. Locking effects in the finite element approximation of elasticity problems. Numer Math, 1992, 62(1):439-463
[7] Babuska I, Suri M. On locking and robustness in the finite element method. SIAM J Numer Anal, 1992,29(5):1261-1293
[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),2017Ge 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