4阶Steklov特征值问题基于降维格式的一种有效的有限元法

An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem

提出了球形区域上4阶Steklov特征值问题基于降维格式的一种有效的有限元方法。首先,通过引入球坐标变换,将原问题化为球坐标系下的等价形式,再利用球调和函数的正交性质进一步化为一系列等价的1维4阶Steklov特征值问题。其次,通过引入极条件和适当的带权Sobolev空间,我们推导了每个1维4阶Steklov特征值问题的弱形式和相应的离散格式,并利用极大极小原理证明了逼近特征值的误差估计。最后,我们给出了一些数值算例,数值结果表明我们的算法是非常有效的。

In this paper,an efficient finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a spherical domain.Firstly,by utilizing the spherical coordinates transformation and the orthogonal property of spherical harmonic function,the original problem is transformed into a series of equivalent one-dimensional fourth-order Steklov eigenvalue problems.Then,we derive the weak form and the corresponding discrete scheme of each one-dimensional fourth-order Steklov eigenvalue problems by introducing the pole conditions and appropriate weighted Sobolev space.Based on the minimax principle,we prove the error estimates of approximation eigenvalues.Finally,we provide some numerical experiments,and the numerical results show the efficiency of our algorithm.