### Abstract

We study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three-dimensional Maxwell's transmission eigenvalue problem. The discretized governing equations by the Nédélec edge element result in a large-scale quadratic eigenvalue problem (QEP) for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form XY^{-1}Z, both of which prevent existing eigenvalue solvers from being efficient. To remedy these difficulties, we rewrite the QEP as a particular nonlinear eigenvalue problem and develop a secant-type iteration, together with an indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method, to sequentially compute the desired positive eigenvalues. Furthermore, we propose a novel method to solve the linear systems in each iteration of LOBPCG. Intensive numerical experiments show that our proposed method is robust, although the desired real eigenvalues are surrounded by complex eigenvalues.

Original language | English |
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Pages (from-to) | A2403-A2423 |

Journal | SIAM Journal on Scientific Computing |

Volume | 37 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

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### Keywords

- LOBPCG
- Maxwell's equations
- Quadratic eigenvalue problems
- Secant-type iteration
- Transmission eigenvalues

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*37*(5), A2403-A2423. https://doi.org/10.1137/15M1018927