Algebraic Multigrid Methods For Higher-Order Finite Element Equations In Two Dimensional Linear Elasticity

In this paper, we discussed to Algebraic Multigrid (AMG) method for higher-order finite element equations in two dimensional linear elasticity. First, we give a commonly used AMG method, and apply it to higher-order finite element equations in two dimensional linear elasticity, and provide the corresponding numerical results. By analyzing the relationship between the linear finite element space and the high order finite element space, we 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. The new AMG method arising from the commonly used AMG method is easy to control the degree of freedom of coarsening grid. Also we give the rigorous theoretical analysis of convergence of the new AMG method (algorithm 4.1 and 4.7). Numerical results show that our AMG method are robust and efficient for solving the high order finite element equations in two dimensional linear elasticity and the convergence rate is a constant independent on the size of the mesh paraments and the discontinuity of material constant (Young Modulus).