电磁场方程在目标识别与成像、隐身与反隐身、遥测遥感、地下目标探测、微波器件的设计等很多领域都有重要的应用。 Nédélec 构造了一系列H(curl)有限元，开启了棱元求解电磁场方程的新时代。本项目的研究目标有两个: 1) 构造一系列H^2(curl)有限元从而使得不高于4阶的电磁场方程均可用协调有限元方法离散; 进而，探究H(curl)和H^2(curl) 元的超收敛性质，同时将保多项式恢复型算子(Polynomial Preserving Recovery)推广到电磁场方程的数值解中。2) 提出极/球几何体上各向同性介质电磁场特征值问题基于降维格式的一种高效高精度的谱Galerkin方法，对于一般区域上各向异性介质电磁场特征值问题提出一种有效的谱元方法。最后, 我们将对提出的数值方法给出理论分析, 进行误差估计，并通过数值试验来验证算法的有效性。

Electromagnetic equations have important applications in many fields such as target recognition and imaging, stealth and anti-stealth, telemetry and remote sensing, underground target detection, microwave device design, etc. Nédélec constructed a class of finite elements belonging to H(curl) , which starts a new era for solving electromagnetic equations by so-called "edge elements". In this project we plan to: 1) construct a series of H^2(curl) finite elements, based on which, all electromagnetic equations with order no more than 4 can be descriticized by conforming finite element methods. Then, we explore superconvergence properties of H(curl) and H^2(curl) elements, and extend Polynomial Preserving Recovery to electromagnetic equations. 2) propose a spectral Galerkin method based on a dimension reduction scheme for the eigenvalue problem of the electromagnetic field in the isotropic medium on the polar/spherical geometry, and a spectral element method for the eigenvalue problem of electromagnetic field in an anisotropic medium on general domains. Finally, we provide error estimation for every numerical method and prove the efficiency of the algorithm by theory and numerical experiments.