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Analysis of HDG methods for Stokes flow


Authors: Bernardo Cockburn, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire and Francisco-Javier Sayas
Journal: Math. Comp. 80 (2011), 723-760
MSC (2010): Primary 65N30, 65M60, 35L65
DOI: https://doi.org/10.1090/S0025-5718-2010-02410-X
Published electronically: September 2, 2010
MathSciNet review: 2772094
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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\ge1$ 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
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
Email: peraire@mit.edu

Francisco-Javier Sayas
Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: sayas002@umn.edu

DOI: https://doi.org/10.1090/S0025-5718-2010-02410-X
Keywords: Stokes flow, mixed methods, discontinuous Galerkin methods, hybridized methods, Lagrange multipliers.
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)
Article copyright: © Copyright 2010 American Mathematical Society

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