An Efficient Preconditioner For Solving Linear Finite Element Equation Of Three-dimensional Linear Elasticity Problems With Neumann Boundary Conditions

Three-dimensional linear elasticity problems with pure Neumann boundary conditions have important application in solid mechanics and computational materials science. Finite element method is among the most effective discretization method to solve three-dimensional linear elasticity problems, but it is still confronted with many difficulties to solve large-scale linear algebraic system efficiently.In this paper, we mainly study the well-posedness and fast solution of a class of linear finite element algebraic systems of three-dimensional linear elasticity problems under pure Neumann boundary condition. First of all, the well-posedness of the continuous variation problem is discussed, and the well-posedness of linear finite element algebraic system is proved by proposing a criterion of how to remove the redundant row in stiffness matrix. Numerical experiments show that the linear finite element error function in L2??? and H1??? norm have saturation error order. Furthermore, three kinds of solving algorithms is discussed for linear finite element algebraic system, which focuses on a Schur complement algorithm and three combined preconditioners based on algebraic multigrid?AMG? method and Gauss Seidel iterative method?GS?. Numerical experiments indicate that the number of iterations and the solution time of Schur complement algorithm and GMRES method based on Combined preconditioner of AMG are more optimal than the commonly used method such as ILU?0?-GMRES. In particular, AMG-GS-?_EE-GMRES method are among the lowest operator complexity and the most effective.