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

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