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 three 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 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) Focusing on the hexahedra domain, investigate Galerkin spectral methods with divergence-free basis functions. Especially, design some curl-orthogonal or curl-quasi-orthogonal basis functions to guarantee the stiff matrix is sparse. This project focuses on the further discussion of high accurate numerical methods for Maxwell and Maxwell transmission 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。本项目拟从如下三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，高精度求解其传输特征值；(2)构造四边形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，实现全局传输特征值高效求解；(3)构造六面体区域满足旋度正交或拟正交的Galerkin谱方法基函数，使刚度矩阵尽量稀疏。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，不仅为复杂区域麦克斯韦传输特征值问题的谱元法奠定基础，而且对沂蒙革命老区数学学科提升、合作交流及开展高水平科研工作具有显著的推动作用。