Volume 1, Issue 1
An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions

Adv. Appl. Math. Mech., 1 (2009), pp. 69-88.

Published online: 2009-01

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• Abstract

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.

• Keywords

Locking phenomenon, algebraic multigrid, higher-order finite element, two-level method, reduced integration.

65N55, 65N22

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@Article{AAMM-1-69, author = {Yingxiong Xiao , and Shi Shu , and Hongmei Zhang , and Ouyang , Yuan}, title = {An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {1}, pages = {69--88}, abstract = {

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.

}, issn = {2075-1354}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aamm/209.html} }
TY - JOUR T1 - An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions AU - Yingxiong Xiao , AU - Shi Shu , AU - Hongmei Zhang , 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.

Yingxiong Xiao, Shi Shu, Hongmei Zhang & Yuan Ouyang. (2020). An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions. Advances in Applied Mathematics and Mechanics. 1 (1). 69-88. doi:
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