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We study a robust and efficient eigensolver for computing the positive dense spectrum of the two-dimensional transmission eigenvalue problem (TEP) which is derived from the Maxwell’s equation with complex media in pseudo-chiral model and the transverse magnetic mode. The discretized governing equations by the N ́ed ́elec edge element result in a large-scale quadratic eigenvalue problem (QEP). We estimate half of the positive eigenvalues of the QEP are on some interval which forms a dense spectrum of the QEP. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Intensive numerical experiments show that our proposed method makes the convergence efficiently and robustly even it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by the non- linear functions which can be applied to estimate the density of the eigenvalues for the TEP.

This article focuses on solving the generalized eigenvalue problems (GEP) arising in the source-free Maxwell equation with magnetoelectric coupling effects that models three-dimensional complex media. The goal is to compute the smallest positive eigenvalues, and the main challenge is that the coefficient matrix in the discrete Maxwell equation is indefinite and degenerate. To overcome this difficulty, we derive a singular value decomposition (SVD) of the discrete single-curl operator and then explicitly express the basis of the invariant subspace corresponding to the nonzero eigenvalues of the GEP. Consequently, we reduce the GEP to a null space free standard eigenvalue problem (NFSEP) that contains only the nonzero (complex) eigenvalues of the GEP and can be solved by the shift-and-invert Arnoldi method without being disturbed by the null space. Furthermore, the basis of the eigendecomposition is chosen carefully so that we can apply fast Fourier transformation (FFT)-based matrix vector multiplication to solve the embedded linear systems efficiently by an iterative method. For chiral and pseudochiral complex media, which are of great interest in magnetoelectric applications, the NFSEP can be further transformed to a null space free generalized eigenvalue problem whose coefficient matrices are Hermitian and Hermitian positive definite (HHPD-NFGEP). This HHPD-NFGEP can be solved by using the invert Lanczos method without shifting. Furthermore, the embedded linear system can be solved efficiently by using the conjugate gradient method without preconditioning and the FFT-based matrix vector multiplications. Numerical results are presented to demonstrate the efficiency of the proposed methods.

This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three dimensional photonic crystals with face centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of $5.184$ million dimension eigenvalue problems within $50$ to $104$ minutes on a workstation with two Intel Quad-C ore Xeon X5687 3.6 GHz CPUs.

The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster and close to the zero eigenvalues. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which is used as a hybrid method that combines with the Jacobi-Davidson and the nonlinear Arnoldi methods to compute the clustering eigenvalues. Numerical results illustrate that the proposed method is robust even for the case of computing many clustering eigenvalues in very large problems.

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 Nedelec 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 method (LOBPCG), 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.

We present an efficient null space free Jacobi–Davidson method to compute the positive eigenvalues of time harmonic Maxwell’s equations. We focus on a class of spatial discretizations that guarantee the existence of discrete vector potentials, such as Yee’s scheme and the edge elements. During the Jacobi–Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in the standard Jacobi–Davidson approach. The computational cost of the transformation from the vector potential to the corrector is negligible. As a consequence, the expanding subspace automatically stays out of the null space and no extra projection step is needed. Numerical evidence confirms that the proposed scheme indeed outperforms the standard and projection-based Jacobi–Davidson methods by a significant margin.

This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter $\gamma$. We show that all the eigenvalues are real and finite for a small chirality $\gamma$. For a critical value $\gamma = \gamma^*$, the GEP has $2 \times 2$ Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when $\gamma$ increases from $\gamma^*$, the $2 \times 2$ Jordan block will first split into a complex conjugate eigenpair, then rapidly collide \tr{with} the real axis and bifurcate into \tr{positive (resonance) and negative eigenvalues with modulus} smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material.