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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Finite element differential forms on cubical meshes
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by Douglas N. Arnold and Gerard Awanou PDF
Math. Comp. 83 (2014), 1551-1570 Request permission

Abstract:

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new $H(\mathrm {curl})$ and $H(\mathrm {div})$ finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.
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Additional Information
  • Douglas N. Arnold
  • Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • MR Author ID: 27240
  • Email: arnold@umn.edu
  • Gerard Awanou
  • Affiliation: Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois at Chicago, Chicago, Illinois 60607-7045
  • MR Author ID: 700956
  • Email: awanou@uic.edu
  • Received by editor(s): April 11, 2012
  • Received by editor(s) in revised form: December 21, 2012
  • Published electronically: October 17, 2013
  • Additional Notes: The work of the first author was supported in part by NSF grant DMS-1115291.
    The work of the second author was supported in part by NSF grant DMS-0811052 and the Sloan Foundation.
  • © Copyright 2013 American Mathematical Society
  • Journal: Math. Comp. 83 (2014), 1551-1570
  • MSC (2010): Primary 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02783-4
  • MathSciNet review: 3194121