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Local and parallel finite element algorithms for magnetohydrodynamic flows with low electromagnetic Reynolds number

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Abstract

In this article, some local and parallel finite element algorithms are proposed and investigated for the magnetohydrodynamic flows with low electromagnetic Reynolds number. For a solution to this problem, it comprises of two main components, the low-frequency components and the high-frequency components. Motivated by this, we obtain the low-frequency components globally via some relatively coarse grid and catch the high-frequency components locally using a fine grid by some local and parallel procedures. Some local a priori estimates that are crucial for our theoretical analysis are derived. The optimal error estimates are rigorously derived and some numerical tests are reported to support our theoretical findings.

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References

  1. Barleon, L., Casal, V., Lenhart, L.: MHD Flow in liquid-metal-cooled blankets. Fusion Eng. Design 14(3-4), 401–412 (1991)

    Article  Google Scholar 

  2. Davidson, P.A.: Magnetohydrodynamics in materials processing. Annu. Rev. Fluid Mech. 31(1), 273–300 (1999)

    Article  Google Scholar 

  3. Davidson, P.A.: An Introduction to magnetohydrodynamics. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  4. Dormy, E., Núñez, M.: Special issue: magnetohydrodynamics in astrophysics and geophysics-introduction. Geo. Astro. Fluid Dynamics 101(3-4), 169–169 (2007)

    Google Scholar 

  5. 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. Eng. 276(7), 287–311 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, G., Zuo, L.: Local and parallel finite element post-processing scheme for the Stokes problem. Comput. Math. Appl. 73(1), 129–140 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Du, G., Zuo, L.: Local and parallel finite element method for the mixed Navier-Stokes/Darcy model with Beavers-Joseph interface conditions. Acta Math. Sci. 37(5), 1331–1347 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Du, G., Zuo, L.: A parallel partition of unity scheme based on two-grid discretizations for the Navier-Stokes problem. J. Sci. Comput. 75(3), 1445–1462 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Du, G., Zuo, L.: Local and parallel finite element methods for the coupled Stokes/Darcy model. Numer. Algo. 87, 1593–1611 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Du, G., Zuo, L., Zhang, Y.: A new local and parallel finite element method for the coupled Stokes-Darcy model. J. Sci. Comput. 90(1), 1–21 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Du, G., Zuo, L.: Local and parallel partition of unity scheme for the mixed Navier-Stokes-Darcy problem. Numer. Algo., pp. 1–16 (2022)

  12. Font, J.A.: General relativistic hydrodynamics and magnetohydrodynamics: hyperbolic systems in relativistic astrophysics, Hyperbolic problems: theory, numerics, applications. Springer, Berlin, Heidelberg, pp. 3–17 (2008)

  13. Gunzburger, M.D., Meir, A.J., Peterson, J.S.: 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 

  14. Gerbeau, J.F.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87(1), 83–111 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gerbeau, J.F., Bris, C.L., Lelièvre, T.: Mathematical methods for the magnetohydrodynamics of liquid metals clarendon press (2006)

  16. He, Y., Xu, J., Zhou, A.: Local and parallel finite element algorithms for the Navier-Stokes problem. J. Comput. Math., pp. 227–238 (2006)

  17. He, Y., Xu, J., Zhou, A.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109(3), 415–434 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. Int. J. Numer. Methods Heat Fluid Flow 25(8), 1912–1923 (2015)

    Article  MATH  Google Scholar 

  19. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Electrodynamics of Continuous Media, Volume 8 of Course of Theoretical Physics, 2nd edn. Butterworth-Heinemann, Oxford (1984)

    Google Scholar 

  20. Lin, T.F, Gilbert, J.B., Kossowsky, R.: Sea-water magnetohydrodynamic propulsion for next-generation undersea vehicles, Pennsylvania state university, State College, PA, United States: applied research laboratory and the nuclear engineering department (1990)

  21. Layton, W., Lenferink, H., Peterson, J.: A two-level newton, finite element algorithm for approximation of electrically conducting incompressible fluid flows. Comput. Math. Appl. 28(2), 21–31 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, Q., Du, G.: Local and parallel finite element methods based on two-grid discretizations for a non-stationary coupled Stokes-Darcy model. Comput. Math. Appl. 113(1), 254–269 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, Q., Du, G.: Local and parallel finite element methods based on two-grid discretizations for a transient coupled Navier-Stokes/Darcy model. J. Sci. Comput. 92(3), 1–35 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  24. Meir, A.J., Schmidt, P.G.: Variational methods for stationary MHD flow under natural interface conditions. Nonlinear Anal. Theory Methods Appl. 26 (4), 659–689 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peterson, J.: On the finite element approximation of incompressible flows of an electrically conducting fluid. Numer. Methods Partial Diff. Equations 4 (1), 57–68 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. Roberts, P.H.: An introduction to magnetohydrodynamics. American Elsevier Pub.Co, New York (1967)

    Google Scholar 

  27. Schonbek, M.E., Schonbek, T.P., Süli, E.: Large-time behaviour of solutions to the magnetohydrodynamics equations. Math. Ann. 304(1), 717–756 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Schmidt, P.G.: A Galerkin method for time-dependent MHD flow with nonideal boundaries. Commun. Appl. Anal. 3(3), 383–398 (1999)

    MathSciNet  MATH  Google Scholar 

  29. Salah, N.B., Soulaimani, A., Habashi, W.G.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190(43), 5867–5892 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Su, H., Feng, X., Huang, P.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Methods Appl. Mech. Eng. 304, 521–545 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Su, H., Feng, X., Zhao, J.: Two-level penalty newton iterative method for the 2d/3d stationary incompressible magnetohydrodynamics equations. J. Sci. Comput. 70, 1144–1179 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shang, Y.: A parallel stabilized finite element method based on the lowest equal-order elements for incompressible flows. Computing 102(1), 65–81 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, X., Du, G., Zuo, L.: A novel local and parallel finite element method for the mixed Navier-Stokes-Darcy problem. Comput. Math. Appl. 90, 73–79 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wu, J., Liu, D., Feng, X., et al.: An efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations. Appl. Math. Comput. 302, 21–33 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69(231), 881–910 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yu, J., Shi, F., Zheng, H.: Local and parallel finite element algorithms based on the partition of unity for the Stokes problem. SIAM J. Sci. Comput. 36(5), C547–C567 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang, G., He, Y.: Decoupled schemes for unsteady MHD equations II: finite element spatial discretization and numerical implementation. Comput. Math. Appl. 69(12), 1390–1406 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, G., He, Y.: Decoupled schemes for unsteady MHD equations. I. time discretization. Numer. Methods Partial Diff. Equations 33(3), 956–973 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, Y., Hou, Y., Shan, L., et al.: Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics. Numer. Methods Partial Diff. Equations 33(5), 1513–1539 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zheng, B., Shang, Y.: Local and parallel finite element algorithms based on domain decomposition for the 2D/3D Stokes equations with damping. Comput. Math. Appl. 103, 82–103 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zheng, B., Shang, Y.: Local and parallel stabilized finite element algorithms based on the lowest equal-order elements for the steady Navier-Stokes equations. Math. Comput. Simulation(MATCOM) 178, 464–484 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zheng, H., Yu, J., Shi, F.: Local and parallel finite element algorithm based on the partition of unity for incompressible flows. J. Sci. Comput. 65 (2), 512–532 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhu, T., Su, H., Feng, X.: Some Uzawa-type finite element iterative methods for the steady incompressible magnetohydrodynamic equations. Appl. Math. Comput. 302, 34–47 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Zuo, L., Du, G.: A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem. Numer. Algo. 77(1), 151–165 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the reviewers for their constructive comments, which allowed for the improvement of the presentation of the results.

Funding

This work is subsidized by NSFC (Nos. 12172202,12001139), the Natural Science Foundation of Shandong Province (No. ZR2021MA063), and the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions (No. 2021KJ037).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Shilin Mi & Guangzhi Du

  2. College of Engineering Physics, Shenzhen Technology University, Shenzhen, Guangdong, 518118, China

    Yao Rong

Authors
  1. Shilin Mi
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  2. Guangzhi Du
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  3. Yao Rong
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Contributions

Shilin Mi: formal analysis, writing, review. Guangzhi Du: conceptualization, methodology, review. Yao Rong: visualization, validation, review. All authors reviewed the manuscript.

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Correspondence to Guangzhi Du.

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Mi, S., Du, G. & Rong, Y. Local and parallel finite element algorithms for magnetohydrodynamic flows with low electromagnetic Reynolds number. Numer Algor 93, 1661–1683 (2023). https://doi.org/10.1007/s11075-022-01484-8

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  • DOI: https://doi.org/10.1007/s11075-022-01484-8

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