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

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.