An Algebraic Multigrid Method For Higher-Order Finite Element Equations In Three Dimensional Linear Elasticity

In this paper, we discuss algebraic multigrid (AMG) methods for higher-order finite element equations in three dimensional linear elasticity. First, we give a commonly used AMG method, and apply it to higher-order finite element equations in three dimensional linear elasticity. Numerical results show that this AMG method is efficient for the linear element, but inefficient for the higher-order elements. By analyzing the relationship between the linear finite element space and the higher-order finite element space, we then obtain a new coarsening algorithm and the corresponding interpolation operator, and apply the AMG method to solving quadratic and cubic Lagrangian finite element discretization systems arising from three dimensional linear elasticity. At the same time we present the rigorous theoretical analysis of convergence for the resulting AMG methods. The new methods overcome one common difficulty in the commonly used AMG method, i.e., the number of coarse grid degrees of freedom is not easy to control. In addition, we present the PCG method for higher-order finite element equations in three dimensional linear elasticity by choosing the inverses of block diagonal matrixes as a preconditioner. Numerical results show that the constructed AMG methods and PCG method are robust and efficient for solving the higher-order finite element equations in three dimensional linear elasticity.