电磁场散射问题 PML 逼近方程的离散系统是一个复杂的对称不定系统，由于该系统的规模一般很大，且高度病态(如curl算子的核空间很大、不同求解区域上方程的性态不同等)，所以为其设计快速算法是非常必要的。迭代两网格法是目前国际上求解偏微分方程离散系统的方法之一。本课题将针对该复杂系统，研究其迭代两网格法，并重点为其中的两个辅助系统设计高效稳健的预条件子。与已有工作相比，我们将面临许多困难点，例如，如何构造高效稳健的预条件子，如何设计合理的数据结构等。解决这些困难，需要发展许多新的方法、理论和技术。这些研究成果将对复杂 Maxwell 方程组的高效数值求解起到积极的重要作用。

The discrete system of PML approximation equation to the electromagnetic wave scattering problem is a very complex, symmetry and indefinite system, which is usually large and higher ill-condition, because the operator curl has a large kernel and the problems on different domain are very different). Hence it is necessary to construct the corresponding fast algorithms for realistic computational ectromagnetism. Iterative two-grid method is popular for solving the discrete systems of partial differential equations. In this project, we plan to study an efficient iterative two-grid method for this complex system and develop efficient and strong preconditioners for two auxiliary systems. Compared to the reported research, we are faced with the difficult which from the construction of efficient preconditioners, development of a reasonable data structure. To solve these difficulties, we need to develop some new methodologies, theories, and techniques. The achievements of this project will be beneficial for developing efficient numerical methods to the complex Maxwell equations.