Consider a linear operator equation $x - Kx = f$, where $f$ is given and $K$ is a Fredholm integral operator with a Green's function type kernel defined on $C[0, 1]$. For $r \geq 1$, we employ the interpolatory projection at $2r + 1$ collocation points (not necessarily Gauss points) onto a space of piecewise polynomials of degree $\leq 2r$ with respect to a uniform partition of $[0, 1]$. Previous researchers have established that the iteration in case of the collocation method improves the order of convergence by projection methods and its variants in the case of smooth kernel with piecewise polynomials of even degree only. In this article, we demonstrate the improvement in order of convergence by modified collocation method when the kernel is of Green's function type.

翻译：考虑线性算子方程 $x - Kx = f$，其中 $f$ 已知，$K$ 为定义在 $C[0, 1]$ 上具有Green函数型核的Fredholm积分算子。对于 $r \geq 1$，我们采用 $2r + 1$ 个配置点（不要求为Gauss点）的插值投影法，投影到关于 $[0, 1]$ 均匀剖分的分段多项式空间（多项式次数 $\leq 2r$）。先前的研究已证明：当核函数光滑且仅使用偶次分段多项式时，配置法的迭代可提升投影法及其变体的收敛阶。本文中，我们证明了当核为Green函数类型时，改进的配置法同样能提升收敛阶。