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

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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.

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