Stability and Convergence of the Integral-Averaged Interpolation Operator Based on $Q_1$-element in $\mathbb{R}^n$

In this paper, we propose an integral-averaged interpolation operator $I_\tau$ in a bounded domain $Ω ⊂ \mathbb{R}^n$ by using $Q_1$-element. The interpolation coefficient is defined by the average integral value of the interpolation function $u$ on the interval formed by the midpoints of the neighboring elements. The operator $I_\tau$ reduces the regularity requirement for the function $u$ while maintaining standard convergence. Moreover, it possesses an important property of $||I_\tau u||_{0,Ω} ≤ ||u||_{0,Ω}.$ We conduct stability analysis and error estimation for the operator $I\tau.$ Finally, we present several numerical examples to test the efficiency and high accuracy of the operator