论文标题
RN中精制的一阶扩展公式：应用于插值和有限元误差估计
A refined first-order expansion formula in Rn: Application to interpolation and finite element error estimates
论文作者
论文摘要
本文的目的是在RN中得出精致的一阶扩展公式，与通常的泰勒公式相比，目标是获得最佳的剩余剩余。对于给定函数，我们得出的公式是通过引入第一个衍生物的线性组合获得的，该组合以$ n+1 $均等的点计算。我们展示了如何将该公式应用于两个重要应用程序：插值误差和有限元素误差估计。在这两种情况下，我们都说明在哪些情况下可以获得误差的重大改善，即使用精制扩展如何减少误差估计的上限。
The aim of this paper is to derive a refined first-order expansion formula in Rn, the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor's formula. For a given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at $n+1$ equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namely how the use of the refined expansion can reduce the upper bound of error estimates.
