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Optimal error estimate of the penalty method for the 2D/3D time-dependent MHD equations

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Abstract

In this article, we mainly consider a first-order decoupling penalty method for the 2D/3D time-dependent incompressible magnetohydrodynamic (MHD) equations in a convex domain. This method applies a penalty term to the constraint “divu = 0,” which allows us to transform the saddle point problem into two small problems to solve. The time discretization is based on the backward Euler scheme. Moreover, we derive the optimal error estimate for the penalty method under semi-discretization with the relationship 𝜖 = O(Δt). Finally, we give abundant of numerical tests to verify the theoretical result and the spatial discretization is based on Lagrange finite element.

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References

  1. Baty, H., Keppens, R., Comte, P.: The two-dimensional magnetohydrodynamic Kelvin-Helmholtz instability: compressibility and large-scale coalescence effects. Phys. Plasmas 10(12), 4661–4674 (2003)

    Article  MathSciNet  Google Scholar 

  2. Becker, R., Hansbo, P.: A simple pressure stabilization method for the Stokes equation. Commun. Numer. Methods Eng. 24(11), 1421–1430 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. Springer, Wiesbaden (1984)

    Book  MATH  Google Scholar 

  4. Chorin, A.: Numerical solution of the Navier-Stokes equations. Math. Comput. 22(104), 745–762 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chorin, A.: On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comput. 23(106), 341–353 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  6. Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1–23 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  7. Deng, J., Si, Z.: A decoupling penalty finite element method for the stationary incompressible magnetohydrodynamics equation. Int. J. Heat Mass Transf. 128, 601–612 (2019)

    Article  Google Scholar 

  8. Donatelli, D.: The artificial compressibility approximation for MHD equations in unbounded domain. J. Hyperbolic Differ. Equ. 10(1), 181–198 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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, 287–311 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Girault, V., Raviart, P.: Finite element methods for Navier-Stokes equations: theory and algorithms. Springer, Berlin (1987)

    MATH  Google Scholar 

  11. Goedbloed, J., Keppens, R., Poedts, S.: Advanced magnetohydrodynamics: with applications to laboratory and astrophysical plasmas. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  12. 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 

  13. He, Y.: Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations. Math. Comput. 74(251), 1201–1216 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J. Numer. Anal. 35(2), 767–801 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. He, Y., Li, J.: A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J. Comput. Appl. Math. 235(3), 708–725 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. John, V.: Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder. Int. J. Numer. Methods Fluids 44(7), 777–788 (2004)

    Article  MATH  Google Scholar 

  18. Lu, X., Lin, P.: Error estimate of the P1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations. Numer. Math. 115(2), 261–287 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Priest, E., Hood, A.: Advances in solar system magnetohydrodynamics. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  20. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36(4), 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shen, J.: On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32(2), 386–403 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

  23. Su, H., Feng, X., Zhao, J.: On two-level Oseen penalty iteration methods for the 2D/3D stationary incompressible magnetohydronamics. J. Sci. Comput. 83(1), 1–30 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Su, H., Mao, S., Feng, X.: Optimal error estimates of penalty based iterative methods for steady incompressible magnetohydrodynamics equations with different viscosities. J. Sci. Comput. 79(2), 1078–1110 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Témam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I). Arch. Ration. Mech. Anal. 32(2), 135–153 (1969)

    Article  MATH  Google Scholar 

  26. Témam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II). Arch. Ration. Mech. Anal. 33(5), 377–385 (1969)

    Article  MATH  Google Scholar 

  27. Yang, J., He, Y.: Stability and error analysis for the first-order Euler implicit/explicit scheme for the 3D MHD equations. Int. J. Comput. Methods 14(2), 1750077 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Yang, J., He, Y., Zhang, G.: On an efficient second order backward difference Newton scheme for MHD system. J. Math. Anal. Appl. 458(1), 676–714 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, X., Zhang, G., He, X.: Convergence analysis of an unconditionally energy stable projection scheme for magneto-hydrodynamic equations. Appl. Numer. Math. 136(1), 235–256 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, Y., Si, Z.: A consistent projection finite element method for the incompressible MHD equations. Appl. Anal. 100(12), 2606–2626 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, G., Chen, C.: Uniformly robust preconditioners for incompressible MHD system. J. Comput. Appl. Math. 379, 112914 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, G., He, X., Yang, X.: A decoupled, linear and unconditionally energy stable scheme with finite element discretizations for magneto-hydrodynamic equations. J. Sci. Comput. 81(3), 1678–1711 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, G., He, X., Yang, X.: Fully decoupled, linear and unconditionally energy stable time discretization scheme for solving the magneto-hydrodynamic equations. J. Comput. Appl. Math. 369, 112636 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, G., He, Y.: Decoupled schemes for unsteady MHD equations. I. Time discretization. Numer. Methods for Partial Differ. Equ. 33(3), 956–973 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, Q., Su, H., Feng, X.: A partitioned finite element scheme based on gauge-Uzawa method for time-dependent MHD equations. Numer. Algoritm. 78(1), 277–295 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are very much indebted to the referees for their constructive suggestions and insightful comments, which greatly improved the original manuscript of this paper.

Funding

This work is partly supported by the NSF of China (No. 12126361, 12126372, 12061076), Tianshan Youth Project of Xinjiang Province (No. 2017Q079), Scientific Research Plan of Universities in the Autonomous Region (No. XJEDU2020I 001), and Key Laboratory Open Project of Xinjiang Province (No. 2020D04002).

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  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Kaiwen Shi, Xinlong Feng & Haiyan Su

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  1. Kaiwen Shi
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Correspondence to Haiyan Su.

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Shi, K., Feng, X. & Su, H. Optimal error estimate of the penalty method for the 2D/3D time-dependent MHD equations. Numer Algor 93, 1337–1371 (2023). https://doi.org/10.1007/s11075-022-01470-0

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