Advertisement

Journal of Scientific Computing

, Volume 31, Issue 1–2, pp 61–73 | Cite as

A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier–Stokes Equations

  • Bernardo CockburnEmail author
  • Guido Kanschat
  • Dominik Schötzau
Article

We present a class of discontinuous Galerkin methods for the incompressible Navier–Stokes equations yielding exactly divergence-free solutions. Exact incompressibility is achieved by using divergence-conforming velocity spaces for the approximation of the velocities. The resulting methods are locally conservative, energy-stable, and optimally convergent. We present a set of numerical tests that confirm these properties. The results of this note naturally expand the work in Cockburn et al. (2005) Math. Comp. 74, 1067–1095.

Keywords

Navier–Stokes equations divergence-free condition discontinuous Galerkin methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold D.N. (1982). An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arnold D.N., Brezzi F., Cockburn B., Marini L.D. (2002). Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baker G.A. (1977). Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31, 45–59zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bassi F., Rebay S. (1997). A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., and Savini, M. (1997). A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In Decuypere, R., and Dibelius, G. (eds.), 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, March 5–7, Technologisch Instituut, pp. 99–108.Google Scholar
  6. 6.
    Bastian P., Rivière B. (2003). Superconvergence and H(div) projection for Galerkin methods. Internat. J. Numer. Methods Fluids 42, 1043–1057zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brezzi, F., Manzini, G., Marini, L. D., Pietra, P., and Russo, A. (1999). Discontinuous finite elements for diffusion problems, in in Atti Convegno in onore di F. Brioschi (Milan 1997), Istituto Lombardo, Accademia di Scienze e Lettere, pp. 197–217.Google Scholar
  8. 8.
    Carrero J., Cockburn B., Schötzau D. (2006). Hybridized, globally divergence-free LDG methods. Part I: The Stokes problem. Math. Comp. 75, 533–563zbMATHGoogle Scholar
  9. 9.
    Castillo P., Cockburn B., Perugia I., Schötzau D. (2000). An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cockburn B., Gopalakrishnan J. (2005). Incompressible finite elements via hybridization. Part I: The Stokes system in two space dimensions. SIAM J. Numer. Anal. 43, 1627–1650zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cockburn B., Gopalakrishnan J. (2005). Incompressible finite elements via hybridization. Part II: The Stokes system in three space dimensions. SIAM J. Numer. Anal. 43, 1651–1672zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cockburn B., Kanschat G., Schötzau D. (2004). Local discontinuous Galerkin methods for the Oseen equations. Math. Comp. 73, 569–593zbMATHGoogle Scholar
  13. 13.
    Cockburn B., Kanschat G., Schötzau D. (2005). The local discontinuous Galerkin methods for linear incompressible flow: A review. Comp. Fluids (Special Issue: Residual distribution schemes, discontinuous Galerkin schemes and Adaptation). 34, 491–506zbMATHGoogle Scholar
  14. 14.
    Cockburn B., Kanschat G., Schötzau D. (2005). A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comp. 74, 1067–1095zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Cockburn B., Kanschat G., Schötzau D., Schwab C. (2002). Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40, 319–343zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cockburn B., Schötzau D., Wang J. (2006). Discontinuous Galerkin methods for incompressible elastic materials. In: Dawson C. (ed), Comput. Methods Appl. Mech. Engrg. 195, 3184–3204Google Scholar
  17. 17.
    Cockburn B., Shu C.-W. (1998). The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kovasznay L. (1948). Laminar flow behind a two-dimensional grid. Proc. Camb. Philos. Soc. 44, 58–62zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Schötzau D., Schwab C., Toselli A. (2003). hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40, 2171–2194zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Bernardo Cockburn
    • 1
    Email author
  • Guido Kanschat
    • 2
  • Dominik Schötzau
    • 3
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany
  3. 3.Mathematics DepartmentUniversity of British ColumbiaVancouverCanada

Personalised recommendations