Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

Tsung Ming Huang, Wei Jen Chang, Yin Liang Huang, Wen Wei Lin, Wei Cheng Wang, Weichung Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yee's scheme as examples, we propose using Krylov-Schur method and Jacobi-Davidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the Krylov-Schur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization.

Original languageEnglish
Pages (from-to)8684-8703
Number of pages20
JournalJournal of Computational Physics
Issue number23
Publication statusPublished - 2010 Nov


  • Eigenvalue problems
  • Fast Fourier transform
  • Harmonic extraction
  • Jacobi-Davidson method
  • Krylov-Schur method
  • Maxwell's equations
  • Preconditioning
  • Three-dimensional photonic crystals

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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