Abstract

Interpolation of a function of one variable with large gradients in the boundary layer region is studied. The problem is that the use of classical polynomial interpolation formulas on a uniform mesh to functions with large gradients can lead to errors of the order of \(O(1)\), despite a small mesh size. An interpolation formula based on fitting to the component that defines the boundary-layer growth of the function is investigated. An error estimate, which depends on the number of interpolation nodes and is uniform over the boundary layer component and its derivatives, is obtained. It is shown how the interpolation formula derived can be used to construct formulas for numerical differentiation and integration and in the two-dimensional case. The corresponding error estimates are obtained.

中文翻译：

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大梯度函数的非多项式插值及其应用

摘要

研究了边界层区域中一个具有大梯度的变量的函数的插值。问题在于，尽管网格尺寸较小，但在均匀网格上使用经典多项式插值公式对具有大梯度的函数可能会导致\（O（1）\）量级的误差。研究了基于拟合函数的插值公式，该分量定义了函数的边界层增长。获得了误差估计值，该误差估计值取决于插值节点的数量，并且在边界层分量及其导数上是均匀的。它显示了如何在二维情况下使用导出的插值公式来构造用于数值微分和积分的公式。获得相应的误差估计。