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Mathematics of Computation

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Finite element differential forms on cubical meshes

Authors: Douglas N. Arnold and Gerard Awanou
Journal: Math. Comp. 83 (2014), 1551-1570
MSC (2010): Primary 65N30
Published electronically: October 17, 2013
MathSciNet review: 3194121
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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

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

Keywords: Mixed finite elements, finite element differential forms, finite element exterior calculus, cubical meshes, cubes
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.
Article copyright: © Copyright 2013 American Mathematical Society