This work is devoted to the numerical computation of complex band structure \(\mathbf {k}=\mathbf {k}(\omega )\in {\mathbb {C}}^3\), with \(\omega \) being positive frequencies, of three dimensional isotropic dispersive or non-dispersive photonic crystals from the perspective of structured quadratic eigenvalue problems (QEPs). Our basic strategy is to fix two degrees of freedom in \(\mathbf {k}\) and to view the remaining one as the eigenvalue of a complex gyroscopic QEP which stems from Maxwell’s equations discretized by Yee’s scheme. We reformulate this gyroscopic QEP into a \(\top \)-palindromic QEP, which is further transformed into a structured generalized eigenvalue problem for which we have established a structure-preserving shift-and-invert Arnoldi algorithm. Moreover, to accelerate the inner iterations of the shift-and-invert Arnoldi algorithm, we propose an efficient preconditioner which makes most of the fast Fourier transforms. The advantage of our method is discussed in detail and corroborated by several numerical results.

这项工作专门用于复杂带结构\（\ mathbf {k} = \ mathbf {k}（\ omega）\在{\ mathbb {C}} ^ 3 \）中的数值计算，其中\（\ omega \）从结构二次特征值问题（QEP）的角度来看，它们是三维各向同性色散或非色散光子晶体的正频率。我们的基本策略是固定\（\ mathbf {k} \）中的两个自由度，并将其余的自由度视为复杂的陀螺仪QEP的特征值，其源于Yee方案离散的Maxwell方程。我们将此陀螺仪QEP重新制定为\（\ top \）-回文QEP，它进一步转化为结构化的广义特征值问题，为此，我们建立了一个保留结构的移位和反转Arnoldi算法。此外，为了加速移位和反转Arnoldi算法的内部迭代，我们提出了一种有效的预处理器，该预处理器可进行大多数快速傅里叶变换。我们的方法的优点已详细讨论，并得到了一些数值结果的证实。