Skip to main content
Log in

Stable finite element methods preserving \(\nabla \cdot \varvec{B}=0\) exactly for MHD models

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss’s law, namely \(\nabla \cdot \varvec{B}=0\). In contrast to most existing approaches that eliminate the electrical field variable \(\varvec{E}\) and give a direct discretization of the magnetic field, our new approach discretizes the electric field \(\varvec{E}\) by Nédélec type edge elements for \(H(\mathrm {curl})\), while the magnetic field \(\varvec{B}\) by Raviart–Thomas type face elements for \(H(\mathrm {div})\). As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for the Picard linearization of the fully nonlinear systems by using the Brezzi theory. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Armero, F., Simo, J.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Badia, S., Codina, R., Planas, R.: On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics. J. Comput. Phys. 234, 399–416 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balsara, D., Kim, J.: A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics. Astrophys. J. 602, 1079–1090 (2004)

    Article  Google Scholar 

  6. Balsara, D., Spicer, D.: A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149, 270–292 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bandaru, V., Boeck, T., Krasnov, D., Schumacher, J.: Numerical computation of liquid metal MHD duct flows at finite magnetic Reynolds number. pamir.sal.lv (1999)

  8. Ba\(\breve{\rm n}\)as, L., Prohl, A.: Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations. Math. Comput. 79(272), 1957–1999 (2010)

  9. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)

  10. Bossavit, A.: Discretization of Electromagnetic Problems: The “Generalized Finite Differences” Approach. Handbook of Numerical Analysis, vol. XIII(04) (2005)

  11. Brackbill, J.: Fluid modeling of magnetized plasmas. Space Plasma Simul. 42, 153–167 (1985)

    Article  Google Scholar 

  12. Brackbill, J.U., Barnes, D.C.: The effect of nonzero \(\nabla \cdot B\) on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 430, 426–430 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. ESAIM. Math. Model. Numer. Anal. 8, 129–151 (1974)

    MathSciNet  MATH  Google Scholar 

  14. Cai, W., Wu, J., Xin, J.: Divergence-free H(div)-conforming hierarchical bases for magnetohydrodynamics (MHD). Commun. Math. Stat. 1, 19–35 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cockburn, B., Li, F., Shu, C.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194(2), 588–610 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Conraths, H.: Eddy current and temperature simulation in thin moving metal strips. Int. J. Numer. Methods Eng. 39, 141–163 (1996)

    Article  MATH  Google Scholar 

  17. Cyr, E., Shadid, J., Tuminaro, R., Pawlowski, R., Chacón, L.: A new approximate block factorization preconditioner for two-dimensional incompressible (reduced) resistive MHD. SIAM J. Sci. Comput. 35(3), 701–730 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dai, W., Woodward, P.: On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows. Astrophys. J. 494(1) (1998)

  19. Davidson, P.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  20. Dedner, A., Kemm, F., Kröner, D., Munz, C., Schnitzer, T., Wessenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175(2), 645–673 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. de Dios, B., Brezzi, F., Marini, L., Xu, J., Zikatanov, L.: A simple preconditioner for a discontinuous Galerkin method for the Stokes problem. J. Sci. Comput. 58(3), 517–547 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Elman, H., Howle, V., Shadid, J., Shuttleworth, R., Tuminaro, R.: Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27, 1651–1668 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Evans, C., Hawley, J.: Simulation of magnetohydrodynamic flows—a constrained transport method. Astrophys. J. 332, 659–677 (1988)

    Article  Google Scholar 

  25. Fey, M., Torrilhon, M.: A constrained transport upwind scheme for divergence-free advection. Hyperbolic Problems: Theory, Numerics, Applications, pp. 529–538 (2003)

  26. Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  27. Guermond, J.L., Minev, P.D.: Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case. Numer. Methods Partial Differ. Equ. 19(6), 709–731 (2003). doi:10.1002/num.10067

  28. Gunzburger, M., Meir, A., Peterson, J.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Math. Comput. 56(194), 523–563 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Helzel, C., Rossmanith, J.A., Taetz, B.: An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations. J. Comput. Phys. 230, 3803–3829 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Hiptmair, R.: Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40(1), 41–65 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hiptmair, R., Heumann, H., Mishra, S., Pagliantini, C.: Discretizing the advection of differential forms. In: ICERM Topical Workshop, Robust Discretization and Fast Solvers for Computable Multi-physics Models. ICERM (2014)

  33. Jardin, S.: Computational Methods in Plasma Physics. CRC Press, Boca Raton (2010)

    Book  MATH  Google Scholar 

  34. Li, F., Shu, C.: Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comput. 22(1–3), 413–442 (2005)

  35. Li, F., Xu, L.: Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations. J. Comput. Phys. 231, 2655–2675 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lin, P., Sala, M., Shadid, J., Tuminaro, R.: Performance of fully-coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport. Int. J. Numer. Methods Eng. 67(9), 208–225 (2006)

    Article  MATH  Google Scholar 

  37. Liu, C.: Energetic variational approaches in complex fluids. In: Multi-Scale Phenomena in Complex Fluids: Modeling, Analysis and Numerical Simulations. World Scientific Publishing Company, Singapore (2009)

  38. Liu, J., Wang, W.: An energy-preserving MAC-Yee scheme for the incompressible MHD equation. J. Comput. Phys. 174(1), 12–37 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, J., Wang, W.: Energy and helicity preserving schemes for hydro- and magnetohydro-dynamics flows with symmetry. J. Comput. Phys. 200, 8–33 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Logg, A., Mardal, K., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  41. Londrillo, P., Zanna, L.: On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method. J. Comput. Phys. 195, 17–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ma, Y., Hu, K., Hu, X., Xu, J.: Robust preconditioners for the magnetohydrodynmaics models (2015). arXiv:1503.02553

  43. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  44. Moreau, R.: Magnetohydrodynamics. Kluwer, Dordrecht (1990)

    Book  MATH  Google Scholar 

  45. Nédélec, J.: Mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  46. Nédélec, J.: A new family of mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 50, 57–81 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  47. Pekmen, B., Tezer-Sezgin, M.: DRBEM solution of incompressible MHD flow with magnetic potential. Comput. Model. Eng. Sci. 96(4), 275–292 (2013)

    MathSciNet  Google Scholar 

  48. Phillips, E., Elman, H., Cyr, E., Shadid, J., Pawlowski, R.: A block preconditioner for an exact penalty formulation for stationary MHD. SIAM J. Sci. Comput. 36(6), 930–951 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Powell, K.: An approximate Riemann solver for magnetohydrodynamics. Upwind and High-Resolution Schemes, pp. 570–583 (1997)

  50. Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. Math. Model. Numer. Anal. 42, 1065–1087 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  51. Raviart, P., Thomas, J.: A mixed finite element method for second order elliptic problems. Lecture Notes Math. 606, 292–315 (1977)

    Article  MATH  Google Scholar 

  52. Rossmanith, J.A.: An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows. SIAM J. Sci. Comput. 28(5), 1766–1797 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  53. Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  54. Schötzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  55. Shadid, J., Cyr, E., Pawlowski, R., Tuminaro, R., Chacón, L., Lin, P.: Initial performance of fully-coupled AMG and approximate block factorization preconditioners for solution of implicit FE resistive MHD. In: Pereira, J., Sequeira, A. (eds.) V Europena Conference on Computational Fluid Dynamics, pp. 1–19 (2010)

  56. Shadid, J., Pawlowski, R., Banks, J., Chacon, L., Lin, P., Tuminaro, R.: Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods. J. Comput. Phys. 229, 7649–7671 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  57. Tóth, G.: The \(\nabla \cdot B= 0\) constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161, 605–652 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tuminaro, R., Tong, C., Shadid, J., Devine, K.: On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz. Commun. Numer. Methods Eng. 18, 383–389 (2002)

    Article  MATH  Google Scholar 

  59. Wiedmer, M.: Finite element approximation for equations of magnetohydrodynamics. Math. Comput. 69, 83–101 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  60. Xu, J.: Fast Auxiliary Space Preconditioning (FASP) Software Package. http://fasp.sourceforge.net/

  61. Ye, X., Hall, C.A.: A discrete divergence-free basis for finite element methods. Numer. Algorithms 16, 365–380 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  62. Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. J. Comput. Phys. 14, 302 (1966)

    MATH  Google Scholar 

  63. Zhang, S.: Bases for C0-P1 divergence-free elements and for C1–P2 finite elements on union jack grids (2012). http://www.math.udel.edu/~szhang/research/p/uj.pdf

Download references

Acknowledgments

The authors would like to thank Long Chen, Xiaozhe Hu, Maximilian Metti, Shuo Zhang, Ludmil Zikatanov for useful discussions and suggestions. And the authors appreciate many valuable suggestions from the anonymous referees.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jinchao Xu.

Additional information

This material is based upon work supported in part by the US Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC-0014400 and by Beijing International Center for Mathematical Research of Peking University, China.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, K., Ma, Y. & Xu, J. Stable finite element methods preserving \(\nabla \cdot \varvec{B}=0\) exactly for MHD models. Numer. Math. 135, 371–396 (2017). https://doi.org/10.1007/s00211-016-0803-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-016-0803-4

Mathematics Subject Classification

Navigation