Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

Journal Article · Sat Nov 20 00:00:00 EST 2010 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2010.08.003· OSTI ID:21418118

Huang, T.-M., E-mail: [e-mail].t ^[1]; Chang, W.-J., E-mail: [e-mail] ^[2]; Huang, Y.-L., E-mail: [e-mail].t ^[3]; Lin, W.-W., E-mail: [e-mail].t ^[4]; Wang, W.-C., E-mail: [e-mail].t ^[5]; Wang Weichung, E-mail: [e-mail].t ^[4]

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1. Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan (China)
2. Graduate Institute of Photonics and Optoelectronic, National Taiwan University, Taipei 106, Taiwan (China)
3. Department of Applied Mathematics, National University of Tainan, Tainan 700, Taiwan (China)
4. Department of Mathematics, National Taiwan University, Taipei 106, Taiwan (China)
5. Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan (China)

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.

• OSTI ID: 21418118
• Journal Information: Journal of Computational Physics, Vol. 229, Issue 23; Other Information: DOI: 10.1016/j.jcp.2010.08.003; PII: S0021-9991(10)00440-7; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICAL METHODS AND COMPUTING
36 MATERIALS SCIENCE
COMPUTERIZED SIMULATION
CRYSTAL STRUCTURE
CRYSTALS
EIGENVALUES
FACTORIZATION
FOURIER TRANSFORMATION
MATHEMATICAL SOLUTIONS
MAXWELL EQUATIONS
OPTIMIZATION
SYMMETRY
THREE-DIMENSIONAL CALCULATIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
INTEGRAL TRANSFORMATIONS
PARTIAL DIFFERENTIAL EQUATIONS
SIMULATION
TRANSFORMATIONS