A refined first-order expansion formula in Rn: Application to interpolation and finite element error estimates

TL;DR

This paper introduces a refined first-order expansion formula in Rn that improves error estimates in interpolation and finite element methods by reducing the upper bounds of errors through a novel linear combination of derivatives.

Contribution

A new refined first-order expansion formula in Rn is developed, enhancing error bounds in interpolation and finite element analysis.

Findings

Significant reduction in error bounds using the refined expansion

Applicable to interpolation error estimation

Applicable to finite element error estimation

Abstract

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.