An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we discuss an algebraic multigrid (AMG) method fornearly incompressible elasticity problems in two-dimensions. First,a two-level method is proposed by analyzing the relationship betweenthe linear finite element space and the quartic finite elementspace. By choosing different smoothers, we obtain two types oftwo-level methods, namely TL-GS and TL-BGS. The theoretical analysisand numerical results show that the convergence rates of TL-GS andTL-BGS are independent of the mesh size and the Young's modulus, andthe convergence of the latter is greatly improved on the order $p$.However, the convergence of both methods still depends on thePoisson's ratio. To fix this, we obtain a coarse level matrix withless rigidity based on selective reduced integration (SRI) methodand get some types of two-level methods by combining differentsmoothers. With the existing AMG method used as a solver on thefirst coarse level, an AMG method can be finally obtained. Numericalresults show that the resulting AMG method has better efficiency fornearly incompressible elasticity problems.