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

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

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 rnultigrid methods, we develop two effective iteration methods：multigrid Uzawa-type mixed finite element methodand nested iteration multigrid Uzawa-type mixed finite element method. Also, we present theconvergent 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.