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Minimal finite element spaces for $ 2m$-th-order partial differential equations in $ R^n$


Authors: Ming Wang and Jinchao Xu
Journal: Math. Comp. 82 (2013), 25-43
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-2012-02611-1
Published electronically: June 8, 2012
MathSciNet review: 2983014
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Abstract: This paper is devoted to a canonical construction of a family of piecewise polynomials with the minimal degree capable of providing a consistent approximation of Sobolev spaces $ H^m$ in $ R^n$ (with $ n\ge m\ge 1$) and also a convergent (nonconforming) finite element space for $ 2m$-th-order elliptic boundary value problems in $ R^n$. For this class of finite element spaces, the geometric shape is $ n$-simplex, the shape function space consists of all polynomials with a degree not greater than $ m$, and the degrees of freedom are given in terms of the integral averages of the normal derivatives of order $ m-k$ on all subsimplexes with the dimension $ n-k$ for $ 1\le k\le m$. This sequence of spaces has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases.

The finite element spaces constructed in this paper constitute the only class of finite element spaces, whether conforming or nonconforming, that are known and proven to be convergent for the approximation of any $ 2m$-th-order elliptic problems in any $ R^n$, such that $ n\ge m\ge 1$. Finite element spaces in this class recover the nonconforming linear elements for Poisson equations ($ m=1$) and the well-known Morley element for biharmonic equations ($ m=2$).


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Additional Information

Ming Wang
Affiliation: Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, China
Email: mwang@math.pku.edu.cn

Jinchao Xu
Affiliation: Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing China –and– Department of Mathematics, Pennsylvania State University
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2012-02611-1
Keywords: Finite element space, minimal degree, conforming, nonconforming, consistent approximation, $2m$-th-order elliptic problem.
Received by editor(s): September 30, 2008
Received by editor(s) in revised form: August 30, 2011
Published electronically: June 8, 2012
Additional Notes: This work was supported by the National Natural Science Foundation of China (10871011)
This work was supported by the National Science Foundation, DMS 0749202 and DMS 0915153
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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