本项目拟在DG框架下提出一种三角形/四面体杂交间断谱元法，并对其开展理论和应用研究。该方法利用四边形到三角形和方块到四面体的变换分别构造三角形和四面体上的谱元基函数。然后借用DG方法处理网格悬点的技术，解决由变换导致的谱元格点在相邻单元公共边或面上可能不匹配的问题。由于离散格式是建立在一阶组上，因此变换雅可比矩阵的奇异性不会给计算离散矩阵带来任何问题。理论分析方面，本项目将研究四面体谱元的L2投影和插值的最优误差估计，并讨论三角形/四面体杂交间断谱元法求解椭圆问题的误差估计。为了在使用高次元时保证计算效率，本项目还将基于求解可分问题的矩阵对角化方法和谱元法的additive Schwarz预处理技术，开展针对三角形/四面体杂交间断谱元离散的预处理子研究。对所设计的预处理子进行严格的理论分析，并用大量数值实验验证所得理论结果。

This program proposes a triangular/tetrahedral hybridizable discontinuous spectral element (THDSE) method and then investigates the theory and application of this high order method. The spectral element basis functions on triangle and tetrahedron are constructed by using mappings from rectangle to triangle and cube to tetrahedron, respectively. The possible inconsistency of the spectral element nodes on the shared edge or face of two adjacent elements induced by the mappings is solved by the DG technique for hanging nodes. The investigated discretization scheme is based on first order system. Thus the singularity of the mapping's Jacobian matrix will not make any problem in the computation of discretization matrices. Optimal error estimate for the L2-projection and interpolation on tetrahedral element is analyzed. Then, L2-error estimates for both 2D and 3D THDSE method for elliptic problems follow. In order to ensure the efficiency, this program also investigates pre-conditioners for both local linear systems and the global linear system using the matrix diagonalization method for separable problems and the methodology of additive Schwarz pre-conditioner for spectral element method. The proposed pre-conditioners shall be studied both theoretically and numerically.