精确零散度的curl协调谱元方法

寻找复杂区域上的计算电磁场问题的高效高精度数值解法是近年来科学计算的一个研究热点。其中，尽可能好地保持物理定律和结构，设计稳定的数值方法是科学计算的一大挑战。.基于此，本项目将建立一类精确零散度的curl协调谱元方法的格式，算法与理论。具体内容如下：（1）研究单区域上的精确零散度的高效谱方法；（2）研究精确零散度的curl协调谱元方法并将其应用到复杂区域上的Maxwell方程组;（3）探讨精确零散度的curl协调谱元方法在其他电磁方程中的应用。与已有的工作相比，这类基于高阶分层基函数的谱元方法，不仅精确满足了Gauss定律，同时兼具了矢量有限元和高阶基函数的优点。 .它们的研究对于拓展谱/谱元方法的基础理论，发展和丰富微分方程的数值解法有着重要的科学意义。

In recent years, the problem of electromagnetic filed is a hot research topic in the area of scientific computing. Its numerical simulations have important values. In this project, our endeavor is devoted to the development of the divergence free and curl conforming spectral element methods. More specifically, 1) we will study the divergence free spectral methods on a single domain；2) we will propose and analyze the divergence free and curl conforming spectral element methods in solving Maxwell equations in complex geometries. 3) we will develop the divergence free and curl conforming spectral element method for other electromagnetic filed problems. Different from the other approaches，the divergence free and curl conforming spectral methods have great significance for keeping the magnetic field divergence free, achieving high accuracy and solving electromagnetic filed problems in complex geometries. The study of the divergence free and curl conforming spectral element methods will enrich and develop the theory and algorithm of spectral/spectral element methods.