On spectral analysis and a novel algorithm for transmission eigenvalue problems. (English) Zbl 1327.65227
The paper provides a spectral analysis and proposes a novel iterative algorithm for the computation of a few positive real eigenvalues and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, the authors derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. Then the QEP is transformed to a parameterized symmetric definite generalized eigenvalue problem (GEP) and a secant-type iteration for solving the resulting GEPs is developed. Moreover, the spectral analysis is carried out for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.
Reviewer: Vit Dolejsi (Praha)
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35P15 | Estimates of eigenvalues in context of PDEs |
Keywords:
transmission eigenvalues; quadratic eigenvalue problems; symmetric positive definite; spectral analysis; secant-type iteration method; eigenfunction; finite element; numerical experimentReferences:
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