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New Analysis of Mixed Finite Element Methods for Incompressible Magnetohydrodynamics

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Abstract

This paper focuses on new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space in cooperation with Navier-Stokes equations and the Nédélec’s edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order Nédélec’s edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order Nédélec’s edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented, which confirm our theoretical analysis.

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References

  1. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21, 823–864 (1998)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numer. 20, 1–119 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Dauge, M.: Neumann and mixed problems on curvilinear polyhedra. Integral Equ. Oper. Theory 15, 227–261 (1992)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Ding, Q., Long, X., Mao, S.: Convergence analysis of Crank-Nicolson extrapolated fully discrete scheme for thermally coupled incompressible magnetohydrodynamic system. Appl. Numer. Math. 157, 522–543 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Ding, Q., Long, X., Mao, S.: Convergence analysis of a fully discrete finite element method for thermally coupled incompressible MHD problems with temperature-dependent coefficients. ESAIM: M2AN 56(3), 969–1005 (2022)

    MathSciNet  MATH  Google Scholar 

  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. Eng. 276, 287–311 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Gao, H., Qiu, W.: A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations. Comput. Methods Appl. Mech. Eng. 346, 982–1001 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Gao, H., Sun, W.: An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg-Landau equations of superconductivity. J. Comput. Phys. 294, 329–345 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Gao, H., Li, B., Sun, W.: Stability and error estimates of fully discrete Galerkin FEMs for nonlinear thermistor equations in non-convex polygons. Numer. Math. 136, 383–409 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Gerbeau, J.F., Le Bris, C., Leliévre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)

    MATH  Google Scholar 

  16. 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. Eng. 199, 2840–2855 (2010)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Guzmán, J., Leykekhman, D., Rossmann, J., Schatz, A.H.: Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods. Numer. Math. 112, 221–243 (2009)

    MathSciNet  MATH  Google Scholar 

  19. He, Y., Zou, J.: A priori estimates and optimal finite element approximation of the MHD flow in smooth domains. ESAIM Math. Model. Numer. Anal. 52, 181–206 (2018)

    MathSciNet  MATH  Google Scholar 

  20. 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)

    MathSciNet  MATH  Google Scholar 

  21. Hiptmair, R., Li, M., Mao, S., Zheng, W.: A fully divergence-free finite element method for magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 28, 659–695 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Hiptmair, R., Pagliantini, C.: Splitting-based structure preserving discretizations for magnetohydrodynamics. SMAI J. Comput. Math. 4, 225–257 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Hu, K., Ma, Y., Xu, J.: Stable finite element methods preserving \(\nabla \cdot {\textbf{B} }=0\) exactly for MHD models. Numer. Math. 135, 371–396 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Hu, K., Xu, J.: Structure-preserving finite element methods for stationary MHD models. Math. Comput. 88, 553–581 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Hughes, W.F., Young, F.J.: The Electromagnetics of Fluids. Wiley, New York (1966)

    Google Scholar 

  26. Jerison, D., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)

    MathSciNet  MATH  Google Scholar 

  27. Jin, D., Ledger, P.D., Gil, A.J.: \(hp\)-finite element solution of coupled stationary magnetohydrodynamics problems including magnetostrictive effects. Comput. Struct. 164, 161–180 (2016)

    Google Scholar 

  28. Lee, J.J., Shannon, S.J., Bui-Thanh, T., Shadid, J.N.: Analysis of an HDG method for linearized incompressible resistive MHD equations. SIAM J. Numer. Anal. 57, 1697–1722 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Li, B., Wang, J., Xu, L.: A convergent linearized lagrange finite element method for the magneto-hydrodynamic equations in two-dimensional nonsmooth and nonconvex domains. SIAM J. Numer. Anal. 58, 430–459 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Li, L., Zheng, W.: A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D. J. Comput. Phys. 351, 254–270 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Meir, A.J., Schmidt, P.G.: Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary. SIAM. J. Numer. Anal. 36, 1304–1332 (1999)

    MathSciNet  MATH  Google Scholar 

  32. Moreau, R.: Magneto-Hydrodynamics. Kluwer Academic Publishers, New York (1990)

    Google Scholar 

  33. Müller, U., Büller, L.: Magnetofluiddynamics in Channels and Containers. Springer, Berlin (2001)

    Google Scholar 

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

    MATH  Google Scholar 

  35. Pagliantini, C.: Computational Magnetohydrodynamics with Discrete Differential Forms, p. 4561. ETH Zürich, Zürich (2016)

    Google Scholar 

  36. Phillips, E.G., Elman, H.C.: A stochastic approach to uncertainty in the equations of MHD kinematics. J. Comput. Phys. 284, 334–350 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Phillips, E.G., Shadid, J.N., Cyr, E.C., Elman, H.C., Pawlowski, R.P.: Block preconditioners for stable mixed nodal and edge finite element representations of incompressible resistive MHD. SIAM J. Sci. Comput. 38(6), B1009–B1031 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Qiu, W., Shi, K.: A mixed DG method and an HDG method for incompressible magnetohydrodynamics. IMA J. Numer. Anal. 40(2), 1356–1389 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Schneebeli, A., Schötzau, D.: Mixed finite elements for incompressible magneto-hydrodynamics,. C. R. Acad. Sci. Paris Ser. I 337(1), 71–74 (2003)

    MathSciNet  MATH  Google Scholar 

  40. Schötzau, D.: Mixed finite element methods for stationary incompressible magnetohydrodynamics. Numer. Math. 96, 315–341 (2004)

    MATH  Google Scholar 

  41. Wathen, M., Greif, Chen: A scalable approximate inverse block preconditioner for an incompressible magnetohydrodynamics model problem. SIAM J. Sci. Comput. 42(1), B57–B79 (2020)

    MathSciNet  MATH  Google Scholar 

  42. Wathen, M., Greif, Chen, Schötzau, D.: Preconditioners for mixed finite element discretizations of incompressible MHD equations. SIAM J. Sci. Comput. 39(6), A2993–A3013 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Zhang, G., 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)

    MathSciNet  MATH  Google Scholar 

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Funding

Weifeng Qiu is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302718). The work of Y. Huang and W. Sun was partially supported by National Natural Science Foundation of China (12231003 and 12071040), Guangdong Provincial Key Laboratory IRADS (2022B1212010006, UIC-R0400001-22) and Guangdong Higher Education Upgrading Plan (UIC-R0400024-21).

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Authors and Affiliations

  1. Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai, 519087, People’s Republic of China

    Yuchen Huang

  2. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China

    Weifeng Qiu

  3. Research Center for Mathematics, Advanced Institute of Natural Science, Beijing Normal University, Zhuhai, People’s Republic of China

    Weiwei Sun

  4. Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science (IRADS), BNU-HKBU United International College, Zhuhai, 519087, People’s Republic of China

    Weiwei Sun

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  1. Yuchen Huang
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YH, WQ and WS have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, method, analysis and writing. All authors certify that this material or similar material has not been and will not be submitted to or published in any other publication.

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Huang, Y., Qiu, W. & Sun, W. New Analysis of Mixed Finite Element Methods for Incompressible Magnetohydrodynamics. J Sci Comput 95, 72 (2023). https://doi.org/10.1007/s10915-023-02189-3

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