本项目致力于推进谱方法在间断问题和非一致介质问题计算中的应用，主要研究内容为：1) 对于偏微分方程间断解问题，充分利用谱方法具有内逼近高精度的优点，设计实施方便的谱元法，建立相应的数值分析理论框架；2)研究非一致介质电磁场间断解问题的高精度谱元法，针对电场和磁场的不同特性，设计合理的逼近空间和内交接面处理技术，获得最优阶误差估计，结合时间方向能量守恒分裂格式和高阶半隐格式，形成计算上述两维和三维问题的有效算法；3)发展三维问题块移动自适应谱元法，结合内部六面体和边界(或内交接面)四面体单元，构造基于逼近函数值和导数值的谱配置法，证明先验误差估计，建立适合块移动谱元法的后验误差指示量，实现分辨率的自适应优化；4）上述方法的并行算法和快速计算，构造单元内部的基本解来分解表示整体逼近解，实现算法的高度并行化和基于Chebyshev点的快速变换。

The aim of this project is to develop the spectral method for discontinuous problems and its applications in computing problems with inhomogeneous media. The main research contents are: 1) For discontinuous problems of partial differential equations, make full use of the high-order accurate inside-approximation properties of spectral methods, design spectral element method with easy implementation, and establish theory framework for numerical analysis; 2) Study high order accurate spectral element methods for discontinuous problems of electromagnetic field with inhomogeneous media, design reasonable approximation spaces and interface conditions with respect to the different characteristics of electric field and magnetic field, obtain the optimal error estimate, and combining with the energy-conserved splitting scheme and high order semi-implicit scheme to formulate efficient algorithm for computing two-dimensional and three-dimensional problems; 3)Develop adaptive block moving spectral element method for three-dimensional problems, combining the hexahedron inside with tetrahedron near boundary or interface, construct the spectral collocation method based on function values and derivative values, prove a priori error estimate, and establish a posteriori error estimator for block moving spectral element method to optimize resolution; 4) Parallel algorithm and fast computation of the above methods, construct the fundamental solutions of inside element to express overall approximation solution, highly parallel algorithms and fast transform based on the Chebyshev points.