Recursive integral method for transmission eigenvalues

Abstract

Recently, a new eigenvalue problem, called the transmission eigenvalue problem, has attracted many researchers. The problem arose in inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing. The problem is numerically challenging because it is non-selfadjoint and nonlinear. In this paper, we propose a recursive integral method for computing transmission eigenvalues from a finite element discretization of the continuous problem. The method, which overcomes some difficulties of existing methods, is based on eigenprojectors of compact operators. It is self-correcting, can separate nearby eigenvalues, and does not require an initial approximation based on some a priori spectral information. These features make the method well suited for the transmission eigenvalue problem whose spectrum is complicated. Numerical examples show that the method is effective and robust.

18 pages, 8 figures

References in corpus (4)

• A Density Matrix-based Algorithm for Solving Eigenvalue Problems
• A spectral projection method for transmission eigenvalues
• A FEAST Algorithm with oblique projection for generalized eigenvalue problems
• A Contour-integral Based Method for Counting the Eigenvalues Inside a Region in the Complex Plane

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• On vanishing and localizing of transmission eigenfunctions near singular points: a numerical study
• An inverse medium problem using Stekloff eigenvalues and a Bayesian approach
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• Finite Element Approximation of Transmission Eigenvalues for Anisotropic Media
• Computation of Transmission Eigenvalues for Elastic Waves
• A lowest order mixed finite element method for the elastic transmission eigenvalue problem
• Spectral Indicator Method for A Non-selfadjoint Steklov Eigenvalue Problem
• A new finite element approach for the Dirichlet eigenvalue problem
• Finite Element Calculation of Photonic Band Structures for Frequency Dependent Materials
• A Multi-level Mixed Element scheme of the two dimensional helmholtz transmission eigenvalue problem
• A Novel Computational Method for Band Structures of Dispersive Photonic Crystals
• A high accuracy nonconforming finite element scheme for Helmholtz transmission eigenvalue problem
• A simple low-degree optimal finite element scheme for the elastic transmission eigenvalue problem
• A Multilevel Spectral Indicator Method for Eigenvalues of Large Non-Hermitian Matrices