﻿ Helmholtz方程透射特征值问题的数值算法 Numerical Solution of Transmission Eigenvalue Problems of Helmholtz Equation

Vol.05 No.04(2016), Article ID:19067,12 pages
10.12677/AAM.2016.54080

Numerical Solution of Transmission Eigenvalue Problems of Helmholtz Equation

Xin Zhou, Tiexiang Li

Department of Mathematics, Southeast University, Nanjing Jiangsu

Received: Nov. 5th, 2016; accepted: Nov. 19th, 2016; published: Nov. 28th, 2016

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. First, we use the continuous finite element method to discrete the transmission eigenvalue problem of Helmholtz equation, and discrete 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. 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.

Keywords:Transmission Eigenvalue, Generalized Eigenvalue Problem, Quadratic Eigenvalue Problem, Linearization, Shift-and-Invert

Helmholtz方程透射特征值问题的数值算法

1. 引言

(1)

2. 透射特征值的离散

2.1. 连续有限元方法

(2)

Table 1. The definition of stiffness matrix and mass matrix

，(3)

2.2. 将广义特征值问题化为二次特征值问题

， (4)

。 (5)

3. 二次特征值问题的求解

。 (6)

。 (7)

， (8)

3.1. 求解线性方程组

(9)

， (10)

，且求解等价于求解。利用分块Gauss消去法有

， (11)

， (12)

。(13)

(14)

3.2. 求解任意位置的特征对

， (15)

，(16)

。 (17)

。 (18)

。 (19)

(20)

4. 数值结果

4.1. 最小的正实透射特征值

4.1.1.为常数

4.1.2.非常数

4.2. 位移求逆

Table 2. When, four smallest real transmission eigenvalues and their error of different domains

Table 3. When, four smallest real transmission eigenvalues and their error of different domains

Table 4. When, four smallest real transmission eigenvalues and their error of different domains

Table 5. When, four smallest real transmission eigenvalues and their error of different domains

Table 6. When, four real transmission eigenvalues near 10 and their error of different domains

5. 总结

Numerical Solution of Transmission Eigenvalue Problems of Helmholtz Equation[J]. 应用数学进展, 2016, 05(04): 683-694. http://dx.doi.org/10.12677/AAM.2016.54080

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