麦克斯韦传输特征值问题零散度约束的高效谱方法研究

The Maxwell transmission eigenvalue problem is becoming a new and challenging topic in the area of inverse scattering theories. During the engineering applications, it is important to design efficient numerical methods for calculating the transmission eigenvalues with high accuracy and devise curl-orthogonal basis functions following the divergence-free conditions. To solve these problems, this project attempts to investigate high effective and accurate numerical methods for Maxwell equations and Maxwell transmission eigenvalue problems from four aspects: (1) Design an efficient Spectral Galerkin method to calculate the eigenvalues of the Maxwell equations using curl-orthogonal or quasi-curl-orthogonal and divergence-free basis functions. (2) For the Maxwell transmission equations on the quadrilateral and hexahedron domain, provide Galerkin spectral methods with exact fully divergence-free conditions to calculate the global transmission eigenvalues, specially device curl-orthogonal or quasi-curl-orthogonal basis functions. (3) Convert Maxwell transmission eigenvalue problems into an equivalent quad-curl equation, and investigate its Galerkin spectral methods with divergence-free basis functions. (4) Code and freely supply the programs for three different domains with spectral methods. This project focuses on the further discussion of high accurate numerical methods for Maxwell.and Maxwell transmission eigenvalue problems, and is a fundamental discussion for spectral element methods on complex domain. Meanwhile, the significant research and academic activities of our school will be enhanced by this project.

麦克斯韦及麦克斯韦传输特征值问题是科学工程领域中极具挑战的热点课题。实际工程模型需要高精度求解其传输特征值，并且构造满足零散度条件的旋度正交或拟正交基函数也尤为困难和重要。本项目拟从如下四方面对麦克斯韦及麦克斯韦传输特征值问题的高效谱方法进行深入研究：(1)采用Galerkin谱方法研究球形区域麦克斯韦及麦克斯韦传输特征值问题，构造满足零散度条件的旋度正交基函数；(2)构造四边形和六面体区域满足零散度条件的Galerkin谱离散格式，构造旋度正交或拟正交基函数；(3)将麦克斯韦传输特征值问题转化为等价的四阶旋度方程，探讨满足零散度条件的基函数构造及其谱方法格式；(4)编写三类区域的谱方法程序，并开源提供。本项目是对麦克斯韦及麦克斯韦传输特征值问题高精度数值模拟的探索，为复杂区域麦克斯韦传输特征值问题的谱元法奠定基础，也对沂蒙革命老区数学学科提升、合作交流及高水平科研工作具有显著的推动作用。