Derivatives of eigenvalues and eigenvectors of matrix functions
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Further Analysis of the Arnoldi Process for Eigenvalue Problems
SIAM Journal on Numerical Analysis
Algorithm 922: A Mixed Finite Element Method for Helmholtz Transmission Eigenvalues
ACM Transactions on Mathematical Software (TOMS)
Error Estimates of the Finite Element Method for Interior Transmission Problems
Journal of Scientific Computing
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Transmission eigenvalues have important applications in inverse scattering theory. They can be used to obtain useful information of the physical properties, such as the index of refraction, of the scattering target. Despite considerable effort devoted to the existence and estimation for the transmission eigenvalues, the numerical treatment is limited. Since the problem is nonstandard, classical finite element methods result in non-Hermitian matrix eigenvalue problems. In this paper, we focus on the computation of a few lowest transmission eigenvalues which are of practical importance. Instead of a non-Hermitian problem, we work on a series of generalized Hermitian problems. We first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem. The roots of these functions are the transmission eigenvalues. Then we apply iterative methods to compute the transmission eigenvalues. We show the convergence of the numerical schemes. The effectiveness of the methods is demonstrated using various numerical examples.