### 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 |
---|---|

Pages (from-to) | A2403-A2423 |

Journal | SIAM Journal on Scientific Computing |

Volume | 37 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

### Fingerprint

### 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

**A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems.** / Hwang, Tsung-Min; Huang, Wei Qiang; Lin, Wen Wei.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 37, no. 5, pp. A2403-A2423. https://doi.org/10.1137/15M1018927

}

TY - JOUR

T1 - A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems

AU - Hwang, Tsung-Min

AU - Huang, Wei Qiang

AU - Lin, Wen Wei

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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-1Z, 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.

AB - 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-1Z, 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.

KW - LOBPCG

KW - Maxwell's equations

KW - Quadratic eigenvalue problems

KW - Secant-type iteration

KW - Transmission eigenvalues

UR - http://www.scopus.com/inward/record.url?scp=84945919789&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945919789&partnerID=8YFLogxK

U2 - 10.1137/15M1018927

DO - 10.1137/15M1018927

M3 - Article

AN - SCOPUS:84945919789

VL - 37

SP - A2403-A2423

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 0036-1445

IS - 5

ER -