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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the consistency of the combinatorial codifferential


Authors: Douglas N. Arnold, Richard S. Falk, Johnny Guzmán and Gantumur Tsogtgerel
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 58A10, 65N30; Secondary 39A12, 57Q55, 58A14
Published electronically: February 26, 2014
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Abstract: In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined. In 1991, Smits proved this to be the case for the combinatorial codifferential applied to $ 1$-forms in two dimensions under the additional assumption that the initial triangulation is refined in a completely regular fashion, by dividing each triangle into four similar triangles. In this paper we extend the result of Smits to arbitrary dimensions, showing that the combinatorial codifferential on $ 1$-forms is consistent if the triangulations are uniform or piecewise uniform in a certain precise sense. We also show that this restriction on the triangulations is needed, giving a counterexample in which a different regular refinement procedure, namely Whitney's standard subdivision, is used. Further, we show by numerical example that for $ 2$-forms in three dimensions, the combinatorial codifferential is not consistent, even for the most regular subdivision process.


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Additional Information

Douglas N. Arnold
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: arnold@umn.edu

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Email: falk@math.rutgers.edu

Johnny Guzmán
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: johnny{\textunderscore}guzman@brown.edu

Gantumur Tsogtgerel
Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 0B9
Email: gantumur@math.mcgill.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06134-5
PII: S 0002-9947(2014)06134-5
Keywords: Consistency, combinatorial codifferential, Whitney form, finite element
Received by editor(s): December 18, 2012
Published electronically: February 26, 2014
Additional Notes: The work of the first author was supported by NSF grant DMS-1115291.
The work of the second author was supported by NSF grant DMS-0910540.
The work of the fourth author was supported by an NSERC Discovery Grant and an FQRNT Nouveaux Chercheurs Grant.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.