An Algebraic Multigrid Method For Compressible And Nearly Incompressible Elasticity Problem In Two Dimensions

In the paper,we discuss the geometric-based algebraic multigrid(AMG) method for compressible and nearly incompressible elasticity problems in two dimensions.First,we study the locking phenomenon of elasticity materials and numerically validate that higherorder (p≥4) conforming finite element can overcome the locking phenomenon.We then propose a two-level method by algebraic approaches for the biquadratic finite element equation of compressible elasticity problems(v≤0.4).By analyzing the relationship between the linear finite element space and the biquadratic finite element space,we obtain a new coarsening algorithm and the corresponding interpolation operator in a purely algebraic way.By choosing different smoothers,such as Ganss-Seidel(GS) and a special block Gauss-Seidel(LBGS) iterations,we obtain two types of two-level methods,namely TL-GS and TL-LBGS methods.Numerical results show that the new methods is more efficient and robust for solving biquadratic finite element discretization systems arising from two-dimensional compressible problem by contrast with the classical AMG which is directly applied to the corresponding discretization systems.At the same time we present the theoretical analysis of convergence for TL-GS method and find that the convergence rate of TL-GS method is independent of the mesh size h,the Young's modulus E and Poisson's ratio v.It is also numerically shown that the convergence rate of TL-LBGS method is greatly improved than TL-GS on the order p.Finally,for nearly incompressible elasticity problems(v→0.5),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.Furthermore,with the existing solver used as a solver on the first coarse level,a geometric-based AMG method can be obtained for biquadratic finite element discretizations as v→0.5.Numerical results have shown that the resulting AMG method has better efficiency and robustness than those methods without using SRI method.