This thesis explores the application of Plane Wave Discontinuous Galerkin (PWDG) methods for the numerical simulation of electromagnetic scattering by periodic structures. Periodic structures play a pivotal role in various engineering and scientific applications, including antenna design, metamaterial characterization, and photonic crystal analysis. Understanding and accurately predicting the scattering behavior of electromagnetic waves from such structures is crucial in optimizing their performance and advancing technological advancements. The thesis commences with an overview of the theoretical foundations of electromagnetic scattering by periodic structures. This theoretical dissertation serves as the basis for formulating the PWDG method within the context of wave equation. The DtN operator is presented and it is used to derive a suitable boundary condition. The numerical implementation of PWDG methods is discussed in detail, emphasizing key aspects such as basis function selection and boundary conditions. The algorithm's efficiency is assessed through numerical experiments. We then present the DtN-PWDG method, which is discussed in detail and is used to derive numerical solutions of the scattering problem. A comparison with the finite element method (FEM) is presented. In conclusion, this thesis demonstrates that PWDG methods are a powerful tool for simulating electromagnetic scattering by periodic structures.

翻译：本论文探索了平面波间断伽辽金（PWDG）方法在周期性结构电磁散射数值模拟中的应用。周期性结构在包括天线设计、超材料表征及光子晶体分析在内的多种工程与科学应用中发挥着关键作用。准确理解并预测此类结构的电磁波散射行为，对于优化其性能及推动技术进步至关重要。论文首先概述了周期性结构电磁散射的理论基础，该理论阐述为在波动方程框架下构建PWDG方法奠定了基础。文中引入DtN算子，并利用其推导出合适的边界条件。详细讨论了PWDG方法的数值实现，重点阐述了基函数选择及边界条件等关键方面。通过数值实验评估了算法的效率。随后，我们详细论述了DtN-PWDG方法，并利用其求解散射问题的数值解。同时，将所得结果与有限元法（FEM）进行了对比分析。总之，本论文证明了PWDG方法是模拟周期性结构电磁散射的强大工具。