We present a framework to solve non-linear eigenvalue problems suitable for a Finite Element discretization. The implementation is based on the open-source finite element software GetDP and the open-source library SLEPc. As template examples, we propose and compare in detail different ways to address the numerical computation of the electromagnetic modes of frequency-dispersive objects. This is a non-linear eigenvalue problem involving a non-Hermitian operator. A classical finite element formulation is derived for five different solutions and solved using algorithms adapted to the large size of the resulting discrete problem. The proposed solutions are applied to the computation of the dispersion relation of a diffraction grating made of a Drude material. The important numerical consequences linked to the presence of sharp corners and sign-changing coefficients are carefully examined. For each method, the convergence of the eigenvalues with respect to the mesh refinement and the shape function order, as well as computation time and memory requirements are investigated. The open-source template model used to obtain the numerical results is provided. Details of the implementation of polynomial and rational eigenvalue problems in GetDP are given in appendix.

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GetDP 和 SLEPc 的非线性特征值问题：频率色散光子开放结构的特征模式计算

我们提出了一个框架来解决适用于有限元离散化的非线性特征值问题。该实现基于开源有限元软件 GetDP 和开源库 SLEPc。作为模板示例，我们提出并详细比较了解决频散物体电磁模式数值计算的不同方法。这是一个涉及非厄米算子的非线性特征值问题。为五种不同的解决方案导出了一个经典的有限元公式，并使用适用于所产生离散问题的大尺寸的算法进行求解。所提出的解决方案应用于计算由 Drude 材料制成的衍射光栅的色散关系。仔细检查了与尖角和符号变化系数的存在相关的重要数值结果。对于每种方法，研究了与网格细化和形状函数阶数相关的特征值的收敛性，以及计算时间和内存要求。提供了用于获得数值结果的开源模板模型。附录中给出了在 GetDP 中实现多项式和有理特征值问题的详细信息。