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An Equal-Order DG Method for the Incompressible Navier-Stokes Equations

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Abstract

We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equal-order approach is ensured by a pressure stabilization term. A simple element-by-element post-processing procedure is used to provide globally divergence-free velocity approximations. For small data, we prove the existence and uniqueness of discrete solutions and carry out an error analysis of the method. A series of numerical results are presented that validate our theoretical findings.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN, 55455, USA

    Bernardo Cockburn

  2. Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX, 77843, USA

    Guido Kanschat

  3. Mathematics Department, University of British Columbia, 121-1984 Mathematics Road, Vancouver, V6T 1Z2, Canada

    Dominik Schötzau

Authors
  1. Bernardo Cockburn
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  2. Guido Kanschat
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  3. Dominik Schötzau
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Corresponding author

Correspondence to Dominik Schötzau.

Additional information

B. Cockburn was supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute.

G. Kanschat was supported in part by NSF through award no. DMS-0713829 and by award no. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).

D. Schötzau was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Cockburn, B., Kanschat, G. & Schötzau, D. An Equal-Order DG Method for the Incompressible Navier-Stokes Equations. J Sci Comput 40, 188–210 (2009). https://doi.org/10.1007/s10915-008-9261-1

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  • DOI: https://doi.org/10.1007/s10915-008-9261-1

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