In this paper, we present an approach to enhance interpolation and approximation error estimates. Based on a previously derived first-order Taylor-like formula, we demonstrate its applicability in improving the $P_1$-interpolation error estimate. Following the same principles, we also develop a novel numerical scheme for the heat equation that yields a better error estimate compared to the classical implicit finite differences scheme.

翻译：本文提出了一种改进插值和逼近误差估计的方法。基于先前推导的一阶泰勒型公式，我们展示了其在改进$P_1$-插值误差估计中的适用性。遵循相同原理，我们还针对热方程开发了一种新的数值格式，相较于经典隐式有限差分格式，该格式能获得更优的误差估计。