Journal of Scientific Computing

, Volume 40, Issue 1–3, pp 281–314 | Cite as

A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics



We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ℘ k 3 −℘ k−1 elements whereas the magnetic part of the equations is approximated by discontinuous ℘ k 3 −℘ k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.


Incompressible magnetohydrodynamics Mixed finite element methods Discontinuous Galerkin methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amestoy, P.R., Duff, I.S., L’Excellent, J.-Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184, 501–520 (2000) MATHCrossRefGoogle Scholar
  2. 2.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Armero, F., Simo, J.C.: Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 131, 41–90 (1996) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Web page (2001).
  6. 6.
    Brenner, S.C.: Poincaré-Friedrichs inequalities for piecewise H 1-functions. SIAM J. Numer. Anal. 41, 306–324 (2003) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. In: Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991) Google Scholar
  8. 8.
    Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection–dominated problems. J. Sci. Comput. 16, 173–261 (2001) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.): Discontinuous Galerkin Methods. Theory, Computation and Applications. Lect. Notes Comput. Sci. Eng., vol. 11. Springer, New York (2000) MATHGoogle Scholar
  10. 10.
    Cockburn, B., Kanschat, G., Schötzau, D.: Local discontinuous Galerkin methods for the Oseen equations. Math. Comput. 73, 569–593 (2004) MATHGoogle Scholar
  11. 11.
    Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74, 1067–1095 (2005) MATHGoogle Scholar
  12. 12.
    Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31, 61–73 (2007) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93, 239–277 (2002) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dauge, M.: Stationary Stokes and Navier–Stokes systems on two- or three-dimensional domains with corners. Part I: Linearized equations. SIAM J. Math. Anal. 20, 74–97 (1989) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gerbeau, J.-F.: Problèmes mathématiques et numériques posés par la modélisation de l’électrolyse de l’aluminium. Ph.D. thesis, Ecole Nationale des Ponts et Chaussées (1998) Google Scholar
  16. 16.
    Gerbeau, J.-F.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gerbeau, J.-F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006) MATHCrossRefGoogle Scholar
  18. 18.
    Giani, S., Hall, E., Houston, P.: AptoFEM Users Manual. Technical report, University of Nottingham (in preparation) Google Scholar
  19. 19.
    Guermond, J.-L., Minev, P.: Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case. Numer. Methods Partial Differ. Equ. 19, 709–731 (2003) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence and uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comput. 56, 523–563 (1991) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hansbo, P., Larson, M.G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191, 1895–1908 (2002) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Texts in Applied Mathematics, vol. 54. Springer, New York (2008) MATHGoogle Scholar
  23. 23.
    Houston, P., Perugia, I., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42, 434–459 (2004) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Houston, P., Perugia, I., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator: Non-stabilized formulation. J. Sci. Comput. 22, 315–346 (2005) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Houston, P., Perugia, I., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case. Math. Model. Numer. Anal. 39, 727–754 (2005) MATHCrossRefGoogle Scholar
  26. 26.
    Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comput. 22–23, 413–442 (2005) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003) MATHCrossRefGoogle Scholar
  28. 28.
    Nédélec, J.C.: Mixed finite element in ℝ3. Numer. Math. 50, 57–81 (1980) CrossRefGoogle Scholar
  29. 29.
    Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. J. Future Gener. Comput. Syst. 20, 475–487 (2004) CrossRefGoogle Scholar
  30. 30.
    Schötzau, D.: Mixed finite element methods for incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Schötzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40, 2171–2194 (2003) MATHCrossRefGoogle Scholar
  32. 32.
    Warburton, T.C., Karniadakis, G.E.: A discontinuous Galerkin method for the viscous MHD equations. J. Comput. Phys. 152, 608–641 (1999) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

Personalised recommendations