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.

翻译：基于一种新的类泰勒公式，我们推导了$W^{1,p}$中改进的插值误差估计。将其与基于标准泰勒公式的经典误差估计以及基于中值定理的相应插值误差估计进行了比较。随后，我们评估了该公式所能带来的精度提升，从而显著降低有限元计算成本。