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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Minimal finite element spaces for $2m$-th-order partial differential equations in $R^n$
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by Ming Wang and Jinchao Xu PDF
Math. Comp. 82 (2013), 25-43 Request permission


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:
  • Jinchao Xu
  • Affiliation: Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing China –and– Department of Mathematics, Pennsylvania State University
  • MR Author ID: 228866
  • Email:
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 82 (2013), 25-43
  • MSC (2010): Primary 65N30
  • DOI:
  • MathSciNet review: 2983014