基于非标准基底的高效谱配置法研究及其应用

谱配置法对于处理变系数或者非线性问题比谱Galerkin方法更有优势。但谱配置法微分矩阵的病态性质严重阻碍了谱配置方法的应用。如何构建新的插值函数系改善微分矩阵条件数、保持数值结果稳定性是科学计算的一大挑战。本项目首先研究无界区域上几类基于非标准型基函数的良态谱配置法和最优误差估计，继而通过求解微分方程构造新的插值基函数系，研究傅里叶化的谱配置法，并探讨在Helmholtz问题和分数阶微分方程求解中的应用，最后建立高维区域上谱配置法的快速算法和数值分析理论。这些新方法具有计算速度快、数值结果稳定、高精度等特点。它们的研究为高效谱元素法提供了有利的工具。

The spectral collocation methods have some advantages over the spectral method using modal basis functions in dealing with variable coefficient and/or nonlinear problems. But ill conditioned differentiation matrices causes severe degradation of expected spectral accuracy. How will we consruct new interpolation basis to better condition numbers and keep stabilization of numerical results? It is a challenge .This project is to investigate well-conditoned collocation methods based on nonstandard basis functions on unbounded domains and their optimal error estimation, to study fourization of spectral collocation methods by constructing new interpolation basis from differential equations, to discuss their applications in Helmholtz problems and fractional differential equations, and to construct fast algorithms on high dimensional domains and numerical analysis. These methods have many advantages, such as faster computing speed, better stability, high accuracy and so on. They provide useful tools to study efficient spectral element methods.