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Nonconforming tetrahedral finite elements for fourth order elliptic equations


Authors: Wang Ming and Jinchao Xu
Journal: Math. Comp. 76 (2007), 1-18
MSC (2000): Primary 65N30
Published electronically: August 1, 2006
MathSciNet review: 2261009
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Abstract: This paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operators in three spatial dimensions. The newly constructed elements include two nonconforming tetrahedral finite elements and one quasi-conforming tetrahedral element. These elements are proved to be convergent for a model biharmonic equation in three dimensions. In particular, the quasi-conforming tetrahedron element is a modified Zienkiewicz element, while the nonmodified Zienkiewicz element (a tetrahedral element of Hermite type) is proved to be divergent on a special grid.


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Additional Information

Wang Ming
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing, People’s Republic of China
Email: mwang@math.pku.edu.cn

Jinchao Xu
Affiliation: The School of Mathematical Sciences, Peking University; Beijing, People’s Republic of China; and Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-06-01889-8
Keywords: Nonconforming finite element, 3-dimension, fourth order elliptic equation, biharmonic
Received by editor(s): October 8, 2004
Received by editor(s) in revised form: September 16, 2005
Published electronically: August 1, 2006
Additional Notes: The work of the first author was supported by the National Natural Science Foundation of China (10571006).
The work of the second author was supported by National Science Foundation DMS-0209479 and DMS-0215392 and the Changjiang Professorship through Peking University
Article copyright: © Copyright 2006 American Mathematical Society