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A locally conservative LDG method for the incompressible Navier-Stokes equations

Authors: Bernardo Cockburn, Guido Kanschat and Dominik Schötzau
Journal: Math. Comp. 74 (2005), 1067-1095
MSC (2000): Primary 65N30
Published electronically: October 5, 2004
MathSciNet review: 2136994
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Abstract: In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in $H(\mathrm {div};\Omega )$ is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.

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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, Minnesota 55455

Guido Kanschat
Affiliation: Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 293/294, 69120 Heidelberg, Germany
MR Author ID: 622524

Dominik Schötzau
Affiliation: Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada

Keywords: Finite element methods, discontinuous Galerkin methods, incompressible Navier-Stokes equations
Received by editor(s): June 10, 2003
Received by editor(s) in revised form: March 12, 2004
Published electronically: October 5, 2004
Additional Notes: The first author was supported in part by the National Science Foundation (Grant DMS-0107609) and by the University of Minnesota Supercomputing Institute
This work was carried out in part while the authors were at the Mathematisches Forschungsinstitut Oberwolfach for the meeting on Discontinuous Galerkin Methods in April 21–27, 2002 and while the second and third authors visited the School of Mathematics, University of Minnesota, in September 2002.
Article copyright: © Copyright 2004 American Mathematical Society