Helmholtz方程透射特征值问题的数值算法

Numerical Solution of Transmission Eigenvalue Problems of Helmholtz Equation

摘要: 本文中我们对Helmholtz方程透射特征值问题提出一种带位移求逆的算法，此算法可以快速有效地求出任意给定的 σ 附近的几个实特征值及对应的特征向量。首先，我们用连续有限元方法对Helmholtz方程透射特征值问题进行离散，并将离散后的广义特征值问题化为一个二次特征值问题，进而对其进行线性化得到一个新的广义特征值问题。这个新的广义特征值问题排除了没有物理意义的零特征值的干扰，保留了所有的非零特征值。我们还利用位移求逆的技术，求得给定的 σ 附近的几个实特征对。我们所提出的算法对透射特征值问题的折射率没有特别的限制，即折射率可为正或负亦或是任意的实函数。最后的数值算例验证了该算法的有效性。

Abstract: In this paper, we put forward a shift-and-invert algorithm to solve the transmission eigenvalue problem of Helmholtz equation, which can quickly and efficiently find out the several eigenvalues and the corresponding eigenvectors near arbitrarily given σ. First, we use the continuous finite element method to discrete the transmission eigenvalue problem of Helmholtz equation, and dis-crete the generalized eigenvalue problem into a quadratic eigenvalue problem, and then a new generalized eigenvalue problem is obtained by linearization. The new generalized eigenvalue problem eliminates the distraction of nonphysical zero eigenvalues, and preserves all the nonzero eigenvalues. Further through the use of shift-and-invert technology, we can quickly and efficiently get several real eigenpairs near given σ. The proposed algorithm has no special restrictions to the refractive index of the transmission eigenvalue problems, that is to say, the refractive index can be positive or negative or positive and negative real numbers. The final numerical example verifies the effectiveness of our algorithm.