TY - JOUR T1 - An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions AU - Xiao , Yingxiong AU - Shu , Shi AU - Zhang , Hongmei AU - Ouyang , Yuan JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 69 EP - 88 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aamm/209.html KW - Locking phenomenon, algebraic multigrid, higher-order finite element, two-level method, reduced integration. AB -

In this paper, we discuss an algebraic multigrid (AMG) method for nearly incompressible elasticity problems in two-dimensions. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and the quartic finite element space. By choosing different smoothers, we obtain two types of two-level methods, namely TL-GS and TL-BGS. The theoretical analysis and numerical results show that the convergence rates of TL-GS and TL-BGS are independent of the mesh size and the Young's modulus, and the convergence of the latter is greatly improved on the order $p$. However, the convergence of both methods still depends on the Poisson's ratio. To fix this, we obtain a coarse level matrix with less rigidity based on selective reduced integration (SRI) method and get some types of two-level methods by combining different smoothers. With the existing AMG method used as a solver on the first coarse level, an AMG method can be finally obtained. Numerical results show that the resulting AMG method has better efficiency for nearly incompressible elasticity problems.