Abstract
We consider the lowest–degree nonconforming finite element methods for the approximation of elliptic problems in high dimensions. The P1–nonconforming polyhedral finite element is introduced for any high dimension. Our finite element is simple and cheap as it is based on the triangulation of domains into parallelotopes, which are combinatorially equivalent to d–dimensional cube, rather than the triangulation of domains into simplices. Our nonconforming element is nonparametric, and on each polytope it contains only linear polynomials, but it is sufficient to give optimal order convergence for second–order elliptic problems.
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Acknowledgements
The research was supported in part by National Research Foundation of Korea (NRF–2017R1A2B3012506 and NRF–2015M3C4A7065662). The author wishes to express his thanks to anonymous referees whose critical comments lead to improve the manuscript substantially.
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Sheen, D. (2020). P1–Nonconforming Polyhedral Finite Elements in High Dimensions. In: de Gier, J., Praeger, C., Tao, T. (eds) 2018 MATRIX Annals. MATRIX Book Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-38230-8_9
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DOI: https://doi.org/10.1007/978-3-030-38230-8_9
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