A new family of mixed finite elements in IR3
Numerische Mathematik
A convergence analysis of Yee's scheme on nonuniform grids
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems
SIAM Journal on Matrix Analysis and Applications
Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions
Mathematics of Computation
Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils
SIAM Journal on Scientific Computing
Mimetic discretizations for Maxwell's equations
Journal of Computational Physics
Thick-restart Lanczos method for electronic structure calculations
Journal of Computational Physics
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Multilevel Method for Mixed Eigenproblems
SIAM Journal on Scientific Computing
A Krylov--Schur Algorithm for Large Eigenproblems
SIAM Journal on Matrix Analysis and Applications
Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Numerical Analysis
Addendum to "A Krylov--Schur Algorithm for Large Eigenproblems"
SIAM Journal on Matrix Analysis and Applications
Convergence Analysis of a Finite Volume Method for Maxwell's Equations in Nonhomogeneous Media
SIAM Journal on Numerical Analysis
A Comparison of Factorization-Free Eigensolvers with Application to Cavity Resonators
ICCS '02 Proceedings of the International Conference on Computational Science-Part III
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
Photonic Crystals: Molding the Flow of Light
Photonic Crystals: Molding the Flow of Light
Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Photonic Crystals, Theory, Applications and Fabrication
Photonic Crystals, Theory, Applications and Fabrication
Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations
Journal of Computational Physics
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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.