Improved $P_1$-interpolation error estimates in $W^{1,p}(]0,1[)$: Application to finite element method

TL;DR

This paper introduces a new Taylor-like formula that improves interpolation error estimates in Sobolev spaces, leading to more accurate finite element computations and reduced costs.

Contribution

It presents a novel Taylor-like formula that enhances interpolation error estimates in $W^{1,p}$, outperforming classical methods.

Findings

Improved error estimates compared to classical Taylor-based estimates.

Significant reduction in finite element computation costs.

Enhanced accuracy in finite element method applications.

Abstract

Based on a new Taylor-like formula, we derived an improved interpolation error estimate in $W^{1,p}$. We compare it with the classical error estimates based on the standard Taylor formula, and also with the corresponding interpolation error estimate, derived from the mean value theorem. We then assess the improvement in accuracy we can get from this formula, leading to a significant reduction in finite element computation costs.