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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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Commuting diagrams for the TNT elements on cubes
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by Bernardo Cockburn and Weifeng Qiu PDF
Math. Comp. 83 (2014), 603-633 Request permission


We present commuting diagrams for the de Rham complex for new elements defined on cubes which use tensor product spaces. The distinctive feature of these elements is that, in sharp contrast with previously known results, they have the TiNiest spaces containing Tensor product spaces of polynomials of degree $k$, hence their acronym TNT. In fact, the local spaces of the TNT elements differ from the standard tensor product spaces by spaces whose dimension is a small number independent of the degree $k$. Such a number is 7 (the number of vertices of the cube minus one) for the space associated with the divergence operator, 18 (the number of faces of the cube times the number of vertices of a face minus one) for the space associated with the curl operator, and 12 (the number of edges of the cube times the number of vertices of an edge minus one) for the space associated with the gradient operator.
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Additional Information
  • Bernardo Cockburn
  • Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • Email:
  • Weifeng Qiu
  • Affiliation: Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455
  • MR Author ID: 845089
  • Email:,
  • Received by editor(s): July 12, 2011
  • Received by editor(s) in revised form: March 5, 2012, March 30, 2012, and June 21, 2012
  • Published electronically: June 18, 2013
  • Additional Notes: The first author was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute.
    The second author gratefully acknowledges the collaboration opportunities provided by IMA (Minneapolis) during their 2010–2012 program. \indent Corresponding author: Weifeng Qiu
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 603-633
  • MSC (2010): Primary 65N30, 65L12
  • DOI:
  • MathSciNet review: 3143686