We consider the Kellogg-Tsan decomposition of the solution to the linear one-dimensional singularly perturbed convection-diffusion problem and improve it by including the solution of the corresponding reduced problem as a component. The upwind scheme on a modified Shishkin-type mesh is used to approximate the unknown component of the decomposition. It is proved that the error is $\mathcal{O}\left(\epsilon {(\ln \epsilon )}^2{N}^{-1}\right)$, where ε is the perturbation parameter and N is the number of mesh steps. The high accuracy of the method is illustrated by numerical examples.

中文翻译：

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奇异摄动对流扩散问题数值方法中改进的 Kellogg-Tsan 解分解

我们考虑线性一维奇异扰动对流扩散问题的解的 Kellogg-Tsan 分解，并通过将相应简化问题的解作为一个组件来改进它。改进的 Shishkin 型网格上的迎风方案用于近似分解的未知分量。证明错误是$\mathcal{哦}\left(\epsilon {(\mathrm{输入}\epsilon )}^2{N}^{-1}\right)$，其中ε是扰动参数，N是网格步长。数值例子说明了该方法的高精度。