特征值问题的组合杂交有限元方法

Combined Hybrid Finite Element Methods for Eigenvalue Problems

摘要:

我们在本文中将组合杂交有限元的思想应用到特征值问题，构造了求解最小特征值问题的一种新型有限元法。首先我们推导了最优误差估计，然后用数值算例验证了理论结果。理论分析和数值算例表明当组合系数 时，我们构造的方法求解最小特征值在最低阶时均能达到二阶精度，并且还能从数值算例中发现对于不同的 ，特征值问题最小值能从左右两个方向趋向于真实值，所以就可以在粗网格上选取最优的 来得到更准确的结果。

Abstract:

In this paper, we apply combined hybrid finite element methods to the eigenvalue problems, and construct a new finite element method for solving the smallest eigenvalue. First, we derive the optimal error estimation, and then we use numerical examples to verify our theoretical results. Both theoretical analyses and numerical results show that for all the combined coefficient , our methods can obtain second order accuracy for solving the smallest eigenvalue when the lowest order finite element spaces are used. From the numerical performance, we can also observe that the numerical solution can approach to the exact eigenvalue from both directions for different , so we can choose the optimal such that better approximations can be obtained on the coarse grid.