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Divergence-conforming HDG methods for Stokes flows

Authors: Bernardo Cockburn and Francisco-Javier Sayas
Journal: Math. Comp. 83 (2014), 1571-1598
MSC (2010): Primary 65M60, 65N30, 35L65
Published electronically: March 19, 2014
MathSciNet review: 3194122
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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

Francisco-Javier Sayas
Affiliation: Department of Mathematical Sciences, University of Delaware, Ewing Hall, Newark, Delaware 19711
MR Author ID: 621885

Keywords: Discontinuous Galerkin methods, hybridization, incompressible fluid flow
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).
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.