本项目旨在建立和分析基于广义Jacobi函数类的高效谱和谱元方法，特别是构造满足物理问题本征性质的非标准基函数，并用于对Navier-Stokes方程和分数阶问题的高精度数值求解。重点在于设计谱和谱元格式严格满足Navier-Stokes方程散度为零的条件，拟合分数阶问题解的本征奇性或渐进性质；同时要考虑离散系统求解的有效性，如：条件数、稀疏结构；本项目也致力于Jacobi函数的谱逼近理论，并应用于所设计的谱和谱元法的稳定性和收敛性分析。

This project aims to establish and analyze efficient spectral methods and spectral-element methods based on generalized Jacobi functions, especially to construct non-standard base functions which such that physical intrinsic properties of underlying problems, and use to solve Navier-Stokes equations and fractional problems high accurately. It is important to design spectral and spectral-element schemes which such the divergence-free property of Navier-Stokes equations exactly, and simulate the intrinsic singularities and asymptotic properties of fractional problems. We also consider the efficiency of solving the discrete system such as condition number and sparse structure. This project also devotes to the approximation theory of generalized Jacobi functions, and use to analyze the convergence and the stability of the proposed spectral and spectral-element schemes.