二维超材料中 Maxwell 方程组高阶 Nedelec 混合有限元超收敛研究

“Cloaking” has received hot attention in electromagnetic field. Effective posteriori error estimators play an essential role in dealing with the design and modeling of cloaking. Superconvergence is one of the way to construct those effective estimators.. For fully discrete Crank-Nicolson scheme of Drude model of Maxwell equations involving metamaterial, we respectively study the superconvergent phenomenon of high order Nedelec mixed finite element solution at the symmetry points of uniform triangular mesh and uniform rectangular mesh. We carry out numerical calculation to confirm our analysis . Furthermore, we apply our results to the design and modeling of cloaking.. Comparing to the existing work, this work has two innovations：Firstly, this work is aimed to obtain the superconvergence of high order Nedelec mixed FEM on the uniform triangular mesh; Secondly, this work is also aimed to obtain the superconvergence of high order Nedelec mixed FEM on the uniform rectangular mesh.. This work is of great significance in the practical application and promote the theoretical knowledge and algorithms for the design of the electromagnetic cloaking in some extent.

电磁波隐身是电磁场备受关注的焦点，而隐身模拟需要有效后验误差估计指示子，超收敛是构造有效估计子的途径之一。. 本课题针对超材料介质中 Maxwell 方程组 Drude 模型全离散 Crank-Nicolson 格式，分别在均匀三角形剖分和均匀矩形剖分上采用高阶 Nedelec 混合限元方法求解，研究限元解在网格对称点(单元中心，边中点，顶点)处的超收敛现象，并且通过数值实验进行验证。进一步地将这些超收敛结果用于电磁波隐身装置的设计与模拟。. 与已有的研究成果相比，本课题有两大创新：第一，本课题拟得到均匀三角形剖分上高阶Nedelec 混合有限元解在网格对称点处超收敛结果；第二，本课题拟得到均匀矩形剖分上高阶Nedelec 混合有限元解在网格对称点处超收敛结果。. 本课题研究具有实际应用价值，在一定程度上推动电磁波隐身装置设计及模拟的理论及算法设计。

从20世纪70年代以来，超收敛是国际研究的热点问题。大多数的研究成果都是针对椭圆和抛物方程的，但是关于麦克斯韦方程的超收敛的结果不是特别多。.该项目主要研究麦克斯韦方程高次棱元的超收敛。该项目针对麦克斯韦方程，采用2次和3次棱有限元方法在矩形网格上求解，在高斯点得到了超收敛结果；