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Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations


Authors: Aycil Cesmelioglu, Bernardo Cockburn and Weifeng Qiu
Journal: Math. Comp. 86 (2017), 1643-1670
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/mcom/3195
Published electronically: November 28, 2016
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Abstract: We present the first a priori error analysis of the hybridizable discontinuous Galerkin method for the approximation of the Navier-Stokes equations proposed in J. Comput. Phys. vol. 230 (2011), pp. 1147-1170. The method is defined on conforming meshes made of simplexes and provides piecewise polynomial approximations of fixed degree $ k$ to each of the components of the velocity gradient, velocity and pressure. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global $ L^2$-norm of the error in each of the above-mentioned variables converges with the optimal order of $ k+1$ for $ k\ge 0$. We also prove a superconvergence property of the velocity which allows us to obtain an elementwise postprocessed approximate velocity, $ H({\rm div})$-conforming and divergence-free, which converges with order $ k+2$ for $ k\ge 1$. In addition, we show that these results only depend on the inverse of the stabilization parameter of the jump of the normal component of the velocity. Thus, if we superpenalize those jumps, these converegence results do hold by assuming that the pressure lies in $ H^1(\Omega )$ only. Moreover, by letting such stabilization parameters go to infinity, we obtain new $ H({\rm div})$-conforming methods with the above-mentioned convergence properties.


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Additional Information

Aycil Cesmelioglu
Affiliation: Department of Mathematics and Statistics, Oakland University, Rochester, Michigan 48309
Email: cesmelio@oakland.edu

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

Weifeng Qiu
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China
Email: weifeqiu@cityu.edu.hk

DOI: https://doi.org/10.1090/mcom/3195
Keywords: Discontinuous Galerkin, hybridization, Navier-Stokes equations, postprocessing, superconvergence
Received by editor(s): February 8, 2015
Received by editor(s) in revised form: January 31, 2016
Published electronically: November 28, 2016
Additional Notes: The second author was supported in part by the National Science Foundation through DMS Grant 1115331.
The third author was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302014). The third author is the corresponding author
Article copyright: © Copyright 2016 American Mathematical Society

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