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

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Bernardo Cockburn, Guido Kanschat and Dominik Schötzau
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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.## References

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## Additional Information

**Bernardo Cockburn**- Affiliation: School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, Minnesota 55455
- Email: cockburn@math.umn.edu
**Guido Kanschat**- Affiliation: Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 293/294, 69120 Heidelberg, Germany
- MR Author ID: 622524
- Email: kanschat@dgfem.org
**Dominik Schötzau**- Affiliation: Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
- Email: schoetzau@math.ubc.ca
- 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. - © Copyright 2004 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 1067-1095 - MSC (2000): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-04-01718-1
- MathSciNet review: 2136994