Abstract
In this paper, a theoretical framework is constructed on how to develop \(C^0\)-nonconforming elements for the fourth order elliptic problem. By using the bubble functions, a simple practical method is presented to construct one tetrahedral \(C^{0}\)-nonconforming element and two cuboid \(C^{0}\)-nonconforming elements for the fourth order elliptic problem in three spacial dimensions. It is also proved that one element is of first order convergence and other two are of second order convergence. From the best knowledge of us, this is the first success in constructing the second-order convergent nonconforming element for the fourth order elliptic problem.
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Acknowledgments
The authors would like to thank the referees for their valuable suggestions. S. Chen and Z. Qiao are grateful for Prof. Xuecheng Tai of University of Bergen. We benefit a lot from the discussion with Prof. Tai during his visit in Hong Kong. H. Chen is partially supported by the Scientific Research Foundation of Graduate School of Zhengzhou University. S. Chen is partially supported by NSFC Grant 11071226. Z. Qiao is partially supported by the Hong Kong RGC Grant PolyU 2017/10P and the Hong Kong Polytechnic University Grants A-PL61 and 1-ZV9Y.
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Chen, H., Chen, S. & Qiao, Z. \(C^{0}\)-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem. Numer. Math. 124, 99–119 (2013). https://doi.org/10.1007/s00211-012-0508-2
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DOI: https://doi.org/10.1007/s00211-012-0508-2