【作者】 张红梅；

【导师】 舒适；

【作者基本信息】 湘潭大学 ， 计算数学， 2007， 硕士

【摘要】 本文主要研究求解三维弹性力学问题的高次有限元方程的代数多层网格(AMG)法。首先，给出了一种常用AMG法，并将其应用于三维线弹性问题高次有限元方程的求解，数值结果表明该方法对高次元的应用效果要差于线性元情形。然后，通过分析线性有限元空间和高次有限元空间基函数之间的内在关系，给出了三维二次和三次有限元空间的网格粗化算法和限制算子的构造方法，进而得到求解线弹性问题高次有限元方程的AMG法，同时对其收敛性给出了严格的理论证明。新的AMG法从本质上克服了粗网格层自由度难以控制等通常AMG法的缺陷。另外，我们还针对线弹性问题给出了基于块对角逆的预条件子的PCG方法。数值实验结果表明，本文所建立的AMG法和PCG方法对求解三维线弹性问题高次有限元方程具有高效性和鲁棒性(robustness)。更多还原

【Abstract】 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.更多还原