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Nonconforming finite element spaces for $ 2m$th order partial differential equations on $ \mathbb{R}^n$ simplicial grids when $ m=n+1$


Authors: Shuonan Wu and Jinchao Xu
Journal: Math. Comp. 88 (2019), 531-551
MSC (2010): Primary 65N30, 65N12
DOI: https://doi.org/10.1090/mcom/3361
Published electronically: May 4, 2018
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Abstract: In this paper, we propose a family of nonconforming finite elements for $ 2m$th order partial differential equations in $ \mathbb{R}^n$ on simplicial grids when $ m=n+1$. This family of nonconforming elements naturally extends the elements proposed by Wang and Xu [Math. Comp. 82 (2013), pp. 25-43], where $ m \leq n$ is required. We prove the unisolvent property by induction on the dimensions using the similarity properties of both shape function spaces and degrees of freedom. The proposed elements have approximability, pass the generalized patch test, and hence converge. We also establish quasi-optimal error estimates in the broken $ H^3$ norm for the 2D nonconforming element. In addition, we propose an $ H^3$ nonconforming finite element that is robust for the sixth order singularly perturbed problems in 2D. These theoretical results are further validated by the numerical tests for the 2D triharmonic problem.


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Additional Information

Shuonan Wu
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: sxw58@psu.edu

Jinchao Xu
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/mcom/3361
Keywords: Finite element method, nonconforming, $2m$th order elliptic problem, triharmonic problem
Received by editor(s): May 30, 2017
Received by editor(s) in revised form: January 9, 2018, and February 5, 2018
Published electronically: May 4, 2018
Additional Notes: This study was partially supported by DOE Grant DE-SC0009249 as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials and by NSF grant DMS-1522615.
Article copyright: © Copyright 2018 American Mathematical Society

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