基于有限带宽基函数的高阶方法

设计拟一致网格和解决高频大波数问题的高精度方法是近年来科学计算的两大主流研究方向。 基于椭圆球面波函数的谱/谱元素方法和p-型的间断谱元素方法则分别是实现拟一致网格高精度方法和解决高频大波数问题的行之有效的途径，但其算法、理论等都亟待发展、丰富和完善。.本项目首先建立基于椭圆球面波函数的谱逼近及插值逼近理论，为拟一致网格的谱/谱元素方法奠定理论基础，特别是最大模的最优意义下的一致估计；探讨拟一致网格的谱元素方法在球面上的浅水波方程求解中的应用。继而研究p-型的间断谱元素方法的算法和实现，发展p-型的间断谱元素逼近求解格式的稳定性和收敛性等数值分析理论；探讨p-型的间断谱元素方法在高频大波数问题求解中的应用。

Recent years,high wave number problems and high order methods which employ quasi-uniform grids have attracted interests of researchers in the area of scientfic computing. The two main approaches are the spectral/spectral element methods using quasi-uniform grids and the p-version discontinuous spectral element methods. .In this project, our endeavor is devoted to the development of the spectral methods with quasi-uniform grids. We will establish spectral approximation theory and interpolation theory based on Prolate Spheroidal Wave Functions, especially uniform estimates in maximun norm with optimal order. And we will investigate the possibility of using the spectral-element method with quasi-uniform grids for solving shallow water equation on the spherical surface. Meanwhile, we will propose and analyze a p-version discontinuous spectral element method for the Helmholtz equations with high wave number.