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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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Analysis of HDG methods for Stokes flow
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by Bernardo Cockburn, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire and Francisco-Javier Sayas PDF
Math. Comp. 80 (2011), 723-760 Request permission

Abstract:

In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree $k$ for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of $k+1$ in $L^2$ for any $k \ge 0$. Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, $\mathbf {H}($div$)$-conforming, and converges with order $k+2$ for $k\ge 1$ and with order $1$ for $k=0$. Numerical experiments are presented which validate the theoretical results.
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Additional Information
  • Bernardo Cockburn
  • Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • Email: cockburn@math.umn.edu
  • Jayadeep Gopalakrishnan
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105
  • MR Author ID: 661361
  • Email: jayg@math.ufl.edu
  • Ngoc Cuong Nguyen
  • Affiliation: Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge Massachusetts 02139
  • Email: cuongng@mit.edu
  • Jaume Peraire
  • Affiliation: Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge Massachusetts 02139
  • MR Author ID: 231923
  • Email: peraire@mit.edu
  • Francisco-Javier Sayas
  • Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain
  • MR Author ID: 621885
  • Email: sayas002@umn.edu
  • Received by editor(s): July 29, 2009
  • Received by editor(s) in revised form: January 5, 2010
  • Published electronically: September 2, 2010
  • Additional Notes: The first author was supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute
    The second author was supported in part by the National Science Foundation under grants DMS-0713833 and SCREMS-0619080
    The third author was supported in part by the Singapore-MIT Alliance
    The fourth author was supported in part by the Singapore-MIT Alliance.
    The fifth author was a Visiting Professor of the School of Mathematics, University of Minnesota, during the development of this work. He was partially supported by MEC/FEDER Project MTM2007–63204 and Gobierno de Aragón (Grupo PDIE)
  • © Copyright 2010 American Mathematical Society
  • Journal: Math. Comp. 80 (2011), 723-760
  • MSC (2010): Primary 65N30, 65M60, 35L65
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02410-X
  • MathSciNet review: 2772094