谱配置法中几类Birkhoff插值问题的收敛性分析

经典的谱配置法求解微分方程时面临着如何处理复杂边界条件和病态的微分矩阵等问题。Birkhoff插值是最近被引入到谱配置法中的一种技术。基于该类插值的谱配置法能精确地满足边界条件，并且离散后的微分矩阵是良态的。鉴于此类新型谱配置法在实际计算中展现的优良性质，本项目将从理论上系统地研究Birkhoff插值问题的收敛性。具体地，针对在求解二阶和三阶微分方程时引入的几类Birkhoff插值问题，根据待插值函数的光滑性，分别研究其代数收敛性、指数收敛性和超收敛性。将建立插值和各阶导数的收敛估计，并寻求插值和各阶导数的超收敛点。分析中的关键是构造插值余项的合适表达式及插值问题在恰当空间中的弱形式，分析中的理论基础是正交多项式的渐进估计和有限元分析技术。本项目的研究不仅能对Birkhoff插值问题的收敛性质建立更深的理解，也为进一步分析基于Birkhoff插值的谱配置法奠定理论基础。

The classical spectral collocation methods suffer from problems such as dealing with complex boundary conditions and ill-conditioned differential matrices when solving differential equations. Birkhoff interpolation has recently been introduced into the spectral collocation methods as a remedy. Based on this kind of interpolation, the boundary conditions are imposed exactly, and the differential matrix is well-conditioned. Given the excellent performance of this new type of spectral collocation method in actual computations, this project will investigate the convergence of the Birkhoff interpolation problem systematically. Specifically, for several kinds of Birkhoff interpolation problems raised in solving second- and third-order differential equations, the algebraic convergence, exponential convergence and super-convergence analysis will be studied according to the smoothness of the interpolated functions. The convergence estimates of the interpolation and the derivatives will be established, and the super-convergence points will be identified. The crucial point in the analysis is to construct the appropriate expressions of the interpolation residuals and the weak form of the interpolation problem in the appropriate space. The theoretical basis for the analysis is the asymptotic estimation of orthogonal polynomials and finite element analysis techniques. The research in this project can not only establish a deeper understanding of the convergence properties of the Birkhoff interpolation problem, but also lay a theoretical foundation for further analysis of the spectral collocation methods based on Birkhoff interpolation.