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Mixed finite element methods for stationary incompressible magneto–hydrodynamics

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Summary.

A new mixed variational formulation of the equations of stationary incompressible magneto–hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nédélec’s elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-size.

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References

  1. Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68, 607–631 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Models Appl. Sci. 21, 823–864 (1998)

    Article  MATH  Google Scholar 

  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. Meth. Appl. Mech. Eng. 131, 41–90 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85, 197–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, vol. 15, Springer, New York, 1991

  6. Ciarlet, P.G.: The finite element method for elliptic problems. North–Holland, Amsterdam, 1978

  7. Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151, 221–276 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93, 239–277 (2002)

    Article  MATH  Google Scholar 

  9. 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)

    MATH  Google Scholar 

  10. Demkowicz, L., Vardapetyan, L.: Modeling of electromagnetic absorption/scattering problems using hp–adaptive finite elements. Comput. Meth. Appl. Mech. Eng. 152, 103–124 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Elman, H.E., Silvester, D.J., Wathen, A.J.: Iterative methods for problems in computational fluid dynamics. Iterative Methods in Scientific Computing, R.H. Chan, T.F. Chan, and G.H. Golub, (eds.), Springer, Singapore, 1997, pp. 271–327

  12. Elman, H.E., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations. Numer. Math. 90, 665–688 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fernandes, P., Gilardi, G.: Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7, 957–991 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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

  15. Gerbeau, J.-F.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations. Springer, New York, 1986

  17. Guermond, J.-L., Minev, P.: Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 2D case. Math. Model. Numer. Anal. 36, 517–536 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Guermond, J.-L., Minev, P.: Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case. Numer. Meth. Part. Diff. Eqns. 19, 709–731 (2003)

    Article  Google Scholar 

  19. 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. Comp. 56, 523–563 (1991)

    MathSciNet  MATH  Google Scholar 

  20. Hiptmair, R.: Multigrid methods for Maxwell’s equations. SIAM J. Numer. Anal. 36, 204–224 (1999)

    Google Scholar 

  21. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica 11, 237–339 (2002)

    Google Scholar 

  22. Hughes, W.F., Young, F.J.: The electromagnetics of fluids. Wiley, New York, 1966

  23. Monk, P.: Finite element methods for Maxwell’s equations. Oxford University Press, New York, 2003

  24. Moreau, R.: Magneto-hydrodynamics. Kluwer Academic Publishers, 1990

  25. Nédélec, J.C.: Mixed finite elements in ℝ3 . Numer. Math. 35, 315–341 (1980)

    Google Scholar 

  26. Nédélec, J.C.: Éléments finis mixtes incompressibles pour l’équation de Stokes dans ℝ3 . Numer. Math. 39, 97–112 (1982)

    MathSciNet  Google Scholar 

  27. Nédélec, J.C.: A new family of mixed finite elements in ℝ3 . Numer. Math. 50, 57–81 (1986)

    MathSciNet  Google Scholar 

  28. Schneebeli, A., Schötzau, D.: Mixed finite elements for incompressible magneto-hydrodynamics. C. R. Acad. Sci. Paris, Série I 337, 71–74 (2003)

    Google Scholar 

  29. Téman, R.: Navier-Stokes equations: Theory and numerical analysis. North-Holland, Amsterdam, 1984

  30. Toselli, A.: Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math. 86, 733–752 (2000)

    MathSciNet  MATH  Google Scholar 

  31. Turek, S.: Efficient solvers for incompressible flow problems. Lecture Notes in Computational Science and Engineering, vol. 6, Springer, 1999

  32. Vardapetyan, L., Demkowicz, L.: hp-adaptive finite elements in electromagnetics. Comput. Meth. Appl. Mech. Eng. 169, 331–344 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Basel, 4051 , Basel, Switzerland

    Dominik Schötzau

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  1. Dominik Schötzau
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Correspondence to Dominik Schötzau.

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Mathematics Subject Classification (2000): 65N30

This work was partially supported by the Swiss National Science Foundation under Project 2100-068126.02.

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Schötzau, D. Mixed finite element methods for stationary incompressible magneto–hydrodynamics. Numer. Math. 96, 771–800 (2004). https://doi.org/10.1007/s00211-003-0487-4

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  • DOI: https://doi.org/10.1007/s00211-003-0487-4

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