超球矩形配置法及其收敛性
The ultraspherical rectangular collocation method and its convergence
AI总结：
本文提出了超球矩形配置（URC）方法，作为Olver & Townsend稀疏超球方法在两点边值问题上的配置实现，证明了其收敛性，并据此启发迭代预处理器设计，同时推广了Driscoll & Hale矩形配置法的收敛性证明。
摘要：
我们开发了超球矩形配置（URC）方法，这是Olver & Townsend针对两点边值问题的稀疏超球方法的一种配置实现。URC方法可证明收敛，实现简单高效，收敛证明启发了迭代方法的预处理器，且配置节点的修改十分直接。当系数函数足够光滑且使用某些超球多项式的根作为配置节点时，收敛定理适用于所有边值问题。我们还调整了Krasnolsel'skii等人的定理以适应我们的设定，证明了Driscoll & Hale的矩形配置方法在受限边界条件类下的收敛性。
英文摘要：
We develop the ultraspherical rectangular collocation (URC) method, a collocation implementation of the sparse ultraspherical method of Olver \& Townsend for two-point boundary-value problems. The URC method is provably convergent, the implementation is simple and efficient, the convergence proof motivates a preconditioner for iterative methods, and the modification of collocation nodes is straightforward. The convergence theorem applies to all boundary-value problems when the coefficient functions are sufficiently smooth and when the roots of certain ultraspherical polynomials are used as collocation nodes. We also adapt a theorem of Krasnolsel'skii et al.~to our setting to prove convergence for the rectangular collocation method of Driscoll \& Hale for a restricted class of boundary conditions.
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