Navier-Stokes(NS)方程的数值解法有着重要科学意义和工程价值，是广为关注的一个前沿和挑战性研究课题。发展稳定高效的数值解法使NS方程能更好解释自然现象。本项目对NS方程设计有能量稳定和时空谱精度等特色的谱元法。借鉴标量辅助变量法，发展能量稳定的数值格式。研究合理空间谱元法，内部用矩形/六面体单元网格划分，边界附近用适当三角形/四面体单元网格，单元上用合理Jacobi Galerkin型谱格式；设计适应时间方向Petrov-Galerkin型谱元法；非线性项用Chebyshev插值等拟谱方法。充分考虑空间满足divergence-free条件和时间方向的基函数；设计适当迭代技术，实现格式长时间计算。本项目获得便于实施的时空谱元法及其数值分析理论，为谱元法在流体模型的应用提供一些技术和理论；培养访问学者科学研究能力，并成为其单位计算数学领域学术骨干，带动他开展高质量研究工作。

The numerical solution for the Navier-Stokes equation is a widely concerned frontier and challenging research topic, which it is of important scientific significance and engineering value. In this project, for the Navier-Stokes equations, we design the spectral element methods with the characteristics, such as, the energy stability and the space-time spectral accuracy, and so on. By using scalar auxiliary variable method, the energy stable numerical scheme is developed. The reasonable space spectral element method is studied, the interior domain is divided by using the rectangular/hexahedral element grids, and the appropriate triangular/tetrahedral element grids are applied to the boundary. The suitable Jacobi-Galerkin spectral scheme is established on the element. The Petrov-Galerkin type element method is designed in time direction. The Chebyshev interpolation method is used in the computations of the nonlinear terms. The divergence-free base functions are fully considered and the base functions adapted to the time direction are also. An appropriate iteration technique is designed, the scheme is implemented in long time behavior calculation. In this project, we obtain the time-space spectral element method, which its numerical implementation is convenient and its numerical analysis theory is given, and some techniques and theories are provided for the application of spectral element method in fluid model. Visiting Scholar’ s the ability of the scientific research is trained, which make him be an academic backbone of Computational mathematics in his department and get him to do quality research.