Skip to main content
SpringerLink
Log in

A Fully Divergence-Free Finite Element Scheme for Stationary Inductionless Magnetohydrodynamic Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we propose and analyze a mixed finite element scheme for stationary inductionless magnetohydrodynamic equations on a general Lipschitz domain. We adopt divergence-conforming elements for the velocity and the current density, discontinuous elements for the pressure and the electric potential, thus the approximations for velocity and current density are exactly divergence-free. The \(\mathbf{H }^{1}\)-continuity of the velocity is enforced by discontinuous Galerkin approach. With this discretization, we prove the well-posedness of the discrete scheme, and derive optimal error estimates of the discrete solutions. In particular, we show that the error estimates for the velocity and the current density are independent of the pressure and the electric potential, and the error estimates for the pressure and the electric potential are also unrelated to each other. Based on this, we propose two coupled iterative methods: Stokes and Oseen iterations. Rigorous analysis of convergence and stability is provided. Finally, some numerical examples are performed to verify the theoretical results and show the effectiveness of the presented methods.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Abdou, M., Ying, A., al. et: On the exploration of innovative concepts for fusion chamber technology. FUSION. ENG. DES. 54(2), 181–247 (2001). https://doi.org/10.1016/S0920-3796(00)00433-6. http://doi.org/10.1016/s0920-3796%2800%2900433-6. Publisher: Elsevier Science

  2. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982). https://doi.org/10.1137/0719052

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02). https://doi.org/10.1137/S0036142901384162

  4. Badia, S., Martín, A.F., Planas, R.: Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem. J. Comput. Phys. 274, 562–591 (2014). https://doi.org/10.1016/j.jcp.2014.06.028

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15. Springer-Verlag, New York (1991). https://doi.org/10.1007/978-1-4612-3172-1

  6. Cockburn, B., Kanschat, G., Schötzau, D.: The local discontinuous Galerkin method for the Oseen equations. Math. Comp. 73(246), 569–593 (2004). https://doi.org/10.1090/S0025-5718-03-01552-7

    Article  MathSciNet  MATH  Google Scholar 

  7. Cockburn, B., Kanschat, G., Schotzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comp. 74(251), 1067–1095 (2005). https://doi.org/10.1090/S0025-5718-04-01718-1

    Article  MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31(1–2), 61–73 (2007). https://doi.org/10.1007/s10915-006-9107-7

    Article  MathSciNet  MATH  Google Scholar 

  9. Davidson, P.A.: An introduction to magnetohydrodynamics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2001). https://doi.org/10.1017/CBO9780511626333

  10. Dong, X., He, Y., Zhang, Y.: Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 276, 287–311 (2014). https://doi.org/10.1016/j.cma.2014.03.022

    Article  MathSciNet  MATH  Google Scholar 

  11. Ervin, V.J., Layton, W.J.: A posteriori error estimation for two level discretizations of flows of electrically conducting, incompressible fluids. Comput. Math. Appl. 31(11), 105–114 (1996). https://doi.org/10.1016/0898-1221(96)00067-3

    Article  MathSciNet  MATH  Google Scholar 

  12. Gerbeau, J.F., Le Bris, C., Lelièvre, T.: Mathematical methods for the magnetohydrodynamics of liquid metals. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2006). https://doi.org/10.1093/acprof:oso/9780198566656.001.0001

  13. Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5. Springer-Verlag, Berlin (1986). https://doi.org/10.1007/978-3-642-61623-5. Theory and algorithms

  14. Girault, V., Rivière, B., Wheeler, M.F.: A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp. 74(249), 53–84 (2005). https://doi.org/10.1090/S0025-5718-04-01652-7

    Article  MathSciNet  MATH  Google Scholar 

  15. Greif, C., Li, D., Schötzau, D., Wei, X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 199(45–48), 2840–2855 (2010). https://doi.org/10.1016/j.cma.2010.05.007

    Article  MathSciNet  MATH  Google Scholar 

  16. He, Y.: Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier-Stokes equations. J. Math. Anal. Appl. 423(2), 1129–1149 (2015). https://doi.org/10.1016/j.jmaa.2014.10.037

    Article  MathSciNet  MATH  Google Scholar 

  17. He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198(15–16), 1351–1359 (2009). https://doi.org/10.1016/j.cma.2008.12.001

    Article  MathSciNet  MATH  Google Scholar 

  18. Hecht, F.: New development in freefem++. J. NUMER. MATH. 20 (2012). https://doi.org/10.1515/jnum-2012-0013

  19. Hiptmair, R., Li, L., Mao, S., Zheng, W.: A fully divergence-free finite element method for magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 28(4), 659–695 (2018). https://doi.org/10.1142/S0218202518500173

    Article  MathSciNet  MATH  Google Scholar 

  20. Hu, K., Ma, Y., Xu, J.: Stable finite element methods preserving \(\nabla \cdot B=0\) exactly for MHD models. Numer. Math. 135(2), 371–396 (2017). https://doi.org/10.1007/s00211-016-0803-4

    Article  MathSciNet  MATH  Google Scholar 

  21. John, V.: Finite element methods for incompressible flow problems, Springer Series in Computational Mathematics, vol. 51. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45750-5

  22. John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017). https://doi.org/10.1137/15M1047696

    Article  MathSciNet  MATH  Google Scholar 

  23. Layton, W., Lenferink, H.W.J., Peterson, J.S.: A two-level Newton, finite element algorithm for approximating electrically conducting incompressible fluid flows. Comput. Math. Appl. 28(5), 21–31 (1994). https://doi.org/10.1016/0898-1221(94)00137-5

    Article  MathSciNet  MATH  Google Scholar 

  24. Layton, W.J., Meir, A.J., Schmidt, P.G.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6(12), 198–210 (1997)

    MathSciNet  MATH  Google Scholar 

  25. Li, L., Ni, M., Zheng, W.: A charge-conservative finite element method for inductionless MHD equations. Part I: Convergence. SIAM J. Sci. Comput. 41(4), B796–B815 (2019). https://doi.org/10.1137/17M1160768

  26. Long, X.: The analysis of finite element method for the inductionless mhd equations. University of Chinese Academy of Sciences, PhD Dissertation pp. 1–123 (2019)

  27. Ni, M.J., Li, J.F.: A consistent and conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part III: On a staggered mesh. J. Comput. Phys. 231(2), 281–298 (2012). https://doi.org/10.1016/j.jcp.2011.08.013

  28. Ni, M.J., Munipalli, R., Huang, P., Morley, N.B., Abdou, M.A.: A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. II. On an arbitrary collocated mesh. J. Comput. Phys. 227(1), 205–228 (2007). https://doi.org/10.1016/j.jcp.2007.07.023

  29. Ni, M.J., Munipalli, R., Morley, N.B., Huang, P., Abdou, M.A.: A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. I. On a rectangular collocated grid system. J. Comput. Phys. 227(1), 174–204 (2007). https://doi.org/10.1016/j.jcp.2007.07.025

  30. Peterson, J.S.: On the finite element approximation of incompressible flows of an electrically conducting fluid. Numer. Methods Partial Differ. Eq. 4(1), 57–68 (1988). https://doi.org/10.1002/num.1690040105

    Article  MathSciNet  MATH  Google Scholar 

  31. Planas, R., Badia, S., Codina, R.: Approximation of the inductionless MHD problem using a stabilized finite element method. J. Comput. Phys. 230(8), 2977–2996 (2011). https://doi.org/10.1016/j.jcp.2010.12.046

    Article  MathSciNet  MATH  Google Scholar 

  32. Su, H., Feng, X., Huang, P.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Methods Appl. Mech. Engrg. 304, 521–545 (2016). https://doi.org/10.1016/j.cma.2016.02.039

    Article  MathSciNet  MATH  Google Scholar 

  33. Yuksel, G., Ingram, R.: Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers. Int. J. Numer. Anal. Model. 10(1), 74–98 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Yuksel, G., Isik, O.R.: Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows. Appl. Math. Model. 39(7), 1889–1898 (2015). https://doi.org/10.1016/j.apm.2014.10.007

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, G.D., He, Y., Yang, D.: Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain. Comput. Math. Appl. 68(7), 770–788 (2014). https://doi.org/10.1016/j.camwa.2014.07.025

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, X., Ding, Q.: Coupled iterative analysis for stationary inductionless magnetohydrodynamic system based on charge-conservative finite element method. J. Sci. Comput. 88(2), 1–32 (2021). https://doi.org/10.1007/s10915-021-01553-5

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for their useful comments and suggestions which have helped to improve the paper greatly.

Author information

Authors and Affiliations

  1. Henan Academy of Big Data, Zhengzhou University, Zhengzhou, 450001, China

    Xiaodi Zhang

  2. LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China

    Xiaorong Wang

  3. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100190, China

    Xiaorong Wang

Authors
  1. Xiaodi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaorong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaorong Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Wang, X. A Fully Divergence-Free Finite Element Scheme for Stationary Inductionless Magnetohydrodynamic Equations. J Sci Comput 90, 70 (2022). https://doi.org/10.1007/s10915-021-01708-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01708-4

Keywords

Mathematics Subject Classification

Search

Navigation