TY - JOUR
T1 - Stability and Convergence of the Integral-Averaged Interpolation Operator Based on $Q_1$-Element in $\mathbb{R}^n$
AU - Liu , Yaru
AU - He , Yinnian
AU - Feng , Xinlong
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 494
EP - 513
PY - 2024
DA - 2024/05
SN - 17
DO - http://doi.org/10.4208/nmtma.OA-2023-0122 
UR - https://global-sci.org/intro/article_detail/nmtma/23109.html
KW - Integral-averaged interpolation operator, $Q_1$-element, stability, convergence.
AB - <p style="text-align: justify;">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.</p>