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Conditions for superconvergence of HDG methods for Stokes flow


Authors: Bernardo Cockburn and Ke Shi
Journal: Math. Comp. 82 (2013), 651-671
MSC (2010): Primary 35L65, 65M60, 65N30
DOI: https://doi.org/10.1090/S0025-5718-2012-02644-5
Published electronically: September 18, 2012
MathSciNet review: 3008833
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Abstract: We provide an a priori error analysis of a wide class of finite element methods for the Stokes equations. The methods are based on the velocity gradient-velocity-pressure formulation of the equations and include new and old mixed and hybridizable discontinuous Galerkin methods. We show how to reduce the error analysis to the verification of some properties of an elementwise-defined projection and of the local spaces defining the methods. We also show that the projection of the errors only depends on the approximation properties of the projection. We then provide sufficient conditions for the superconvergence of the projection of the error in the approximate velocity. We give many examples of these methods and show how to systematically construct them from similar methods for the diffusion equation.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Ke Shi
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: shixx075@math.umn.edu

DOI: https://doi.org/10.1090/S0025-5718-2012-02644-5
Received by editor(s): August 22, 2011
Received by editor(s) in revised form: August 26, 2011
Published electronically: September 18, 2012
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.
Article copyright: © Copyright 2012 American Mathematical Society

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