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Volume 17, Issue 2
Stability and Convergence of the Integral-Averaged Interpolation Operator Based on $Q_1$-Element in $\mathbb{R}^n$

Yaru Liu, Yinnian He & Xinlong Feng

DOI: 10.4208/nmtma.OA-2023-0122

Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 494-513.

Published online: 2024-05

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  • Abstract

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.

  • Keywords

Integral-averaged interpolation operator, $Q_1$-element, stability, convergence.

  • AMS Subject Headings

41A05, 97N50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-17-494, author = {Liu , YaruHe , Yinnian and Feng , Xinlong}, title = {Stability and Convergence of the Integral-Averaged Interpolation Operator Based on $Q_1$-Element in $\mathbb{R}^n$}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {2}, pages = {494--513}, abstract = {

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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2023-0122 }, url = {http://global-sci.org/intro/article_detail/nmtma/23109.html} }
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 -

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.

Liu , YaruHe , Yinnian and Feng , Xinlong. (2024). Stability and Convergence of the Integral-Averaged Interpolation Operator Based on $Q_1$-Element in $\mathbb{R}^n$. Numerical Mathematics: Theory, Methods and Applications. 17 (2). 494-513. doi:10.4208/nmtma.OA-2023-0122
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