Two mixed spectral-element methods (MSEMs) are proposed to solve Maxwell's eigenvalue problems with Bloch (Floquet) periodic and open resonators to remove the dc spurious modes present in the traditional numerical methods. The first MSEM combines the variational form of Gauss's law to make the electric field satisfy the divergence-free condition. This method does not need to modify the Arnoldi algorithm for the eigenvalue problems, but the number of degrees of freedom (DOFs) is larger than that of the traditional spectral-element method (SEM). The second MSEM, as a generalization of the tree-cotree technique, can eliminate the dc spurious modes by imposing the constrained equations on the Ritz vectors in the Arnoldi algorithm without increasing the DOF from the traditional SEM. We expand the electric field in terms of the edge basis functions associated with the cotree edges and the gradient of the nodal basis functions so that the discrete gradient matrix G̅ consists of the identity matrix G̅t and the zero matrix G̅c, making the matrix inversion of G̅t trivial. Finally, in the numerical experiments, we compare the computational costs of the MSEMs and finite-element method (FEM) in COMSOL to illustrate the high efficiency of the MSEMs.

中文翻译：

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使用布洛赫周期和开路谐振器求解 3-D 麦克斯韦特征值问题的混合谱元方法


提出了两种混合谱元方法（MSEM）来解决带有布洛赫（Floquet）周期和开放谐振器的麦克斯韦特征值问题，以消除传统数值方法中存在的直流杂散模式。第一个MSEM结合了高斯定律的变分形式，使电场满足无发散条件。该方法不需要对特征值问题的Arnoldi算法进行修改，但自由度数(DOF)比传统谱元法(SEM)要大。第二个MSEM是树-共树技术的推广，可以通过在Arnoldi算法中对Ritz向量施加约束方程来消除直流杂散模式，而无需增加传统SEM的DOF。我们根据与余树边相关的边基函数和节点基函数的梯度来扩展电场，使得离散梯度矩阵G̅由单位矩阵G̅t和零矩阵G̅c组成，使得G̅t的矩阵求逆琐碎的。最后，在数值实验中，我们比较了 MSEM 和 COMSOL 中的有限元方法 (FEM) 的计算成本，以说明 MSEM 的高效率。