球域上传输特征值问题的一种有效的谱逼近

为了求解球形区域上的内部传输特征值问题，提出一种有效的谱逼近方法。首先，定义一种乘积型Sobolev空间，利用单位球上一类正交多项式构造相应的逼近空间。其次，通过引入一个辅助函数，将原问题转化为一个等价的四阶混合格式，并推导出该四阶混合格式的变分形式及其离散格式。然后，利用投影算子的逼近性质和Babu2ka-Osborn理论，证明逼近解的误差估计。最后，详细地描述算法的实现过程，并通过一些数值算例验证了算法的收敛性和高精度。

An efficient spectral approximation for the transmission eigenvalue problem in spherical domains

In order to solve the interior transmission eigenvalue problems in a spherical region, an effective spectral approximation method is proposed. Firstly, a product type Sobolev space is defined, and the corresponding approximation space is constructed by using a class of orthogonal polynomials on the unit sphere. Then, by introducing an auxiliary function, the original problem is transformed into an equivalent fourth-order mixed scheme, and the variational form and discrete form of the fourth-order mixed scheme are derived. Moreover, by using the approximation property of the projection operator and Babuška-Osborn theory, the error estimation of approximation solution is proved. Finally, the implementation process of the algorithm is described in detail, and some numerical examples are given to verify the convergence and high accuracy of the algorithm.