### Abstract

The band structures of three-dimensional photonic crystals can be determined numerically by solving a sequence of generalized eigenvalue problems. However, not all of the spectral structures of these eigenvalue problems are well-understood, and not all of these eigenvalue problems can be solved efficiently. This article focuses on the eigenvalue problems corresponding to wave vectors that are close to the center of the Brillouin zone of a three dimensional, simple cubic photonic crystal. For these eigenvalue problems, there are (i) many zero eigenvalues, (ii) a couple of near-zero eigenvalues, and (iii) several larger eigenvalues. As the desired eigenvalues are the smallest positive eigenvalues, these particular spectral structures prevent regular eigenvalue solvers from efficiently computing the desired eigenvalues. We study these eigenvalue problems from the perspective of both theory and computation. On the theoretical side, the structures of the null spaces are analyzed to explicitly determine the number of zero eigenvalues of the target eigenvalue problems. On the computational side, the Krylov-Schur and Jacobi-Davidson methods are used to compute the smallest, positive, interior eigenvalues that are of interest. Intensive numerical experiments disclose how the shift values, conditioning numbers, and initial vectors affect the performance of the tested eigenvalue solvers and suggest the most efficient eigenvalue solvers.

Original language | English |
---|---|

Pages (from-to) | 529-551 |

Number of pages | 23 |

Journal | Journal of Scientific Computing |

Volume | 55 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 Jan 1 |

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### Keywords

- Eigenvalue problems
- Jacobi-Davidson method
- Krylov-Schur method
- Maxwell's equations
- Numerical linear algebra
- Preconditioners
- Spectrum analysis
- Threedimensional photonic crystals

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Numerical Analysis
- Engineering(all)
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Scientific Computing*,

*55*(3), 529-551. https://doi.org/10.1007/s10915-012-9646-z

**Computing extremal eigenvalues for three-dimensional photonic crystals with wave vectors near the brillouin zone center.** / Huang, Tsung Ming; Kuo, Yueh Cheng; Wang, Weichung.

Research output: Contribution to journal › Article

*Journal of Scientific Computing*, vol. 55, no. 3, pp. 529-551. https://doi.org/10.1007/s10915-012-9646-z

}

TY - JOUR

T1 - Computing extremal eigenvalues for three-dimensional photonic crystals with wave vectors near the brillouin zone center

AU - Huang, Tsung Ming

AU - Kuo, Yueh Cheng

AU - Wang, Weichung

PY - 2013/1/1

Y1 - 2013/1/1

N2 - The band structures of three-dimensional photonic crystals can be determined numerically by solving a sequence of generalized eigenvalue problems. However, not all of the spectral structures of these eigenvalue problems are well-understood, and not all of these eigenvalue problems can be solved efficiently. This article focuses on the eigenvalue problems corresponding to wave vectors that are close to the center of the Brillouin zone of a three dimensional, simple cubic photonic crystal. For these eigenvalue problems, there are (i) many zero eigenvalues, (ii) a couple of near-zero eigenvalues, and (iii) several larger eigenvalues. As the desired eigenvalues are the smallest positive eigenvalues, these particular spectral structures prevent regular eigenvalue solvers from efficiently computing the desired eigenvalues. We study these eigenvalue problems from the perspective of both theory and computation. On the theoretical side, the structures of the null spaces are analyzed to explicitly determine the number of zero eigenvalues of the target eigenvalue problems. On the computational side, the Krylov-Schur and Jacobi-Davidson methods are used to compute the smallest, positive, interior eigenvalues that are of interest. Intensive numerical experiments disclose how the shift values, conditioning numbers, and initial vectors affect the performance of the tested eigenvalue solvers and suggest the most efficient eigenvalue solvers.

AB - The band structures of three-dimensional photonic crystals can be determined numerically by solving a sequence of generalized eigenvalue problems. However, not all of the spectral structures of these eigenvalue problems are well-understood, and not all of these eigenvalue problems can be solved efficiently. This article focuses on the eigenvalue problems corresponding to wave vectors that are close to the center of the Brillouin zone of a three dimensional, simple cubic photonic crystal. For these eigenvalue problems, there are (i) many zero eigenvalues, (ii) a couple of near-zero eigenvalues, and (iii) several larger eigenvalues. As the desired eigenvalues are the smallest positive eigenvalues, these particular spectral structures prevent regular eigenvalue solvers from efficiently computing the desired eigenvalues. We study these eigenvalue problems from the perspective of both theory and computation. On the theoretical side, the structures of the null spaces are analyzed to explicitly determine the number of zero eigenvalues of the target eigenvalue problems. On the computational side, the Krylov-Schur and Jacobi-Davidson methods are used to compute the smallest, positive, interior eigenvalues that are of interest. Intensive numerical experiments disclose how the shift values, conditioning numbers, and initial vectors affect the performance of the tested eigenvalue solvers and suggest the most efficient eigenvalue solvers.

KW - Eigenvalue problems

KW - Jacobi-Davidson method

KW - Krylov-Schur method

KW - Maxwell's equations

KW - Numerical linear algebra

KW - Preconditioners

KW - Spectrum analysis

KW - Threedimensional photonic crystals

UR - http://www.scopus.com/inward/record.url?scp=84893670112&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893670112&partnerID=8YFLogxK

U2 - 10.1007/s10915-012-9646-z

DO - 10.1007/s10915-012-9646-z

M3 - Article

AN - SCOPUS:84893670112

VL - 55

SP - 529

EP - 551

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 3

ER -