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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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Divergence-conforming HDG methods for Stokes flows
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by Bernardo Cockburn and Francisco-Javier Sayas PDF
Math. Comp. 83 (2014), 1571-1598 Request permission

Abstract:

In this paper, we show that by sending the normal stabilization function to infinity in the hybridizable discontinuous Galerkin methods previously proposed in [Comput. Methods Appl. Mech. Engrg. 199 (2010), 582–597], for Stokes flows, a new class of divergence-conforming methods is obtained which maintains the convergence properties of the original methods. Thus, all the components of the approximate solution, which use polynomial spaces of degree $k$, converge with the optimal order of $k+1$ in $L^2$ for any $k \ge 0$. Moreover, the postprocessed velocity approximation is also divergence-conforming, exactly divergence-free and converges with order $k+2$ for $k\ge 1$ and with order $1$ for $k=0$. The novelty of the analysis is that it proceeds by taking the limit when the normal stabilization goes to infinity in the error estimates recently obtained in [Math. Comp., 80 (2011) 723–760].
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Additional Information
  • Bernardo Cockburn
  • Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
  • Email: cockburn@math.umn.edu
  • Francisco-Javier Sayas
  • Affiliation: Department of Mathematical Sciences, University of Delaware, Ewing Hall, Newark, Delaware 19711
  • MR Author ID: 621885
  • Email: fjsayas@udel.edu
  • Received by editor(s): July 25, 2011
  • Received by editor(s) in revised form: December 31, 2012
  • Published electronically: March 19, 2014
  • 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 was a Visiting Professor of the School of Mathematics, University of Minnesota, during the development of this work, and was partially supported by the National Science Foundation (Grant DMS 1216356).
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 1571-1598
  • MSC (2010): Primary 65M60, 65N30, 35L65
  • DOI: https://doi.org/10.1090/S0025-5718-2014-02802-0
  • MathSciNet review: 3194122