非局部数学模型已经被证实，它们在模拟许多物理现象时比传统的模型更加贴切. 非局部算子的主要困难在于:(i)算子是全局的，具有历史和整体的依赖性;(ii)奇异核函数或权函数往往致使解具有局部奇性.谱方法在处理这类问题具有一定的优势. 本项目面向若干带奇异和非局部算子的模型，构造几类非标准基底，建立相应的正交逼近和插值逼近理论，提出高效的谱Galerkin方法和谱配置法，研究离散代数线性系统的数值稳定性和算法的收敛性. 它们的研究拓展了谱方法的应用领域，为高效谱元素法提供了有利的工具.

Mathematical models with nonlocal operators have proven to be of great value and superior to conventional models in modeling many physical phenomena. Some common and interwoven difficulties lie in that the non-local operators have (i) inherent history and global dependence; and (ii) singular kernel/weight functions, so the underlying solutions are usually locally singular. The spectral methods are proven to be more attractive for such nonlocal singular problems. This project is to construct some non-standard basis functions, to establish the basic approximation results, to propose efficient spectral methods for some problems with nonlocal operators and singular solutions. In addition, we will study numerical stability of the corresponding discrete systems and the convergence rate of the proposed methods. They provide useful tools to study efficient spectral element methods.