Band structure calculation of frequency dependent photonic crystals has important applications. The associated eigenvalue problem is nonlinear and the development of convergent numerical methods is challenging. In this paper, we formulate the band structure problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Lagrange finite elements are used to discretize the operators. The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions. Then a spectral indicator method is developed to practically compute the eigenvalues. Numerical examples are presented to validate the theory and show the effectiveness of the proposed method.

频率相关的光子晶体的能带结构计算具有重要的应用。相关的特征值问题是非线性的，并且收敛数值方法的发展具有挑战性。在本文中，我们将能带结构问题表述为指数为零的全纯Fredholm算子函数的特征值问题。拉格朗日有限元用于离散算子。利用全纯算子函数的抽象逼近理论证明了特征值的收敛性。然后，开发了一种频谱指示符方法来实际计算特征值。数值算例验证了该理论的有效性，并证明了该方法的有效性。