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All issues Volume 57 / No 5 (September-October 2023) ESAIM: M2AN, 57 5 (2023) 2907-2930 References
Open Access
Issue
ESAIM: M2AN
Volume 57, Number 5, September-October 2023
Page(s) 2907 - 2930
DOI https://doi.org/10.1051/m2an/2023061
Published online 19 September 2023
  1. D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1–155. [Google Scholar]
  2. D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229–246. [Google Scholar]
  3. D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. Vol. 44 of Springer Series in Computational Mathematics, Springer, Heidelberg (2013). [CrossRef] [Google Scholar]
  4. J.U. Brackbill and D.C. Barnes, The effect of nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35 (1980) 426–430. [NASA ADS] [CrossRef] [Google Scholar]
  5. B. Cockburn, G. Kanschat and D. Schötzau, A nonote on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31 (2007) 61–73. [CrossRef] [MathSciNet] [Google Scholar]
  6. M. Costabel and M. Dauge, Singularities of Maxwell’s equations on polyhedral domains, in Analysis, Numerics and Applications of Differential and Integral Equations (Stuttgart, 1996). Vol. 379 of Pitman Res. NONOTEs Math. Ser. Longman, Harlow (1996) 69–76. [Google Scholar]
  7. M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000) 221–276. [Google Scholar]
  8. W. Dai and P.R. Woodward, On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamical flows. Astrophys. J. 494 (1998) 317–335. [CrossRef] [Google Scholar]
  9. P.A. Davidson, An Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2001). [CrossRef] [Google Scholar]
  10. Q. Ding, X. Long and S. Mao, Convergence analysis of Crank-Nicolson extrapolated fully discrete scheme for thermally coupled incompressible magnetohydrodynamic system. Appl. Numer. Math. 157 (2020) 522–543. [CrossRef] [MathSciNet] [Google Scholar]
  11. H. Duan, S. Li, R.C.E. Tan and W. Zheng, A delta-regularization finite element method for a double cnourl problem with divergence-free constraint. SIAM J. Numer. Anal. 50 (2012) 3208–3230. [CrossRef] [MathSciNet] [Google Scholar]
  12. K.J. Galvin, A. Linke, L.G. Rebholz and N.E. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. Comput. Methods Appl. Mech. Eng. 237240 (2012) 166–176. [CrossRef] [Google Scholar]
  13. H. Gao and W. Qiu, A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations. Comput. Methods Appl. Mech. Eng. 346 (2019) 982–1001. [Google Scholar]
  14. J.-F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2006). [Google Scholar]
  15. V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). [CrossRef] [Google Scholar]
  16. C. Greif, D. Li, D. Schötzau and X. Wei, A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 199 (2010) 2840–2855. [Google Scholar]
  17. M.D. Gunzburger, A.J. Meir and J.S. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Math. Comput. 56 (1991) 523–563. [CrossRef] [Google Scholar]
  18. J. He, K. Hu and J. Xu, Generalized Gaffney inequality and discrete compactness for discrete differential forms. Numer. Math. 143 (2019) 781–795. [CrossRef] [MathSciNet] [Google Scholar]
  19. Y. He, Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J. Numer. Anal. 35 (2015) 767–801. [Google Scholar]
  20. F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. [Google Scholar]
  21. R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237–339. [Google Scholar]
  22. R. Hiptmair, L. Li, S. Mao and W. Zheng, A fully divergence-free finite element method for magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 28 (2018) 659–695. [Google Scholar]
  23. K. Hu, Y. Ma and J. Xu, Stable finite element methods preserving ∇ ・ B = 0 exactly for MHD models. Numer. Math. 135 (2017) 371–396. [CrossRef] [MathSciNet] [Google Scholar]
  24. V. John, Finite Element Methods for Incompressible Flow Problems. Vol. 51 of Springer Series in Computational Mathematics, Springer, Cham (2016). [CrossRef] [Google Scholar]
  25. V. John, A. Linke, C. Merdon, M. Neilan and L.G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59 (2017) 492–544. [CrossRef] [MathSciNet] [Google Scholar]
  26. L. Li, Finite element methods and fast solvers for incompressible magetohydrodynamic systems. Ph.D. thesis, AMSS, Chinese Academy of Sciences (2018). [Google Scholar]
  27. L. Li, M. Ni and W. Zheng, A charge-conservative finite element method for inductionless MHD equations. Part I: convergence. SIAM J. Sci. Comput. 41 (2019) B796–B815. [CrossRef] [Google Scholar]
  28. L. Li, M. Ni and W. Zheng, A charge-conservative finite element method for inductionless MHD equations. Part II: a robust solver. SIAM J. Sci. Comput. 41 (2019) B816–B842. [CrossRef] [Google Scholar]
  29. L. Li, D. Zhang and W. Zheng, A constrained transport divergence-free finite element method for incompressible MHD equations. J. Comput. Phys. 428 (2021) 109980. [CrossRef] [Google Scholar]
  30. P. Monk, Finite Element Methods. [Google Scholar]
  31. M.-J. Ni and J.-F. Li, 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 (2012) 281–298. [CrossRef] [MathSciNet] [Google Scholar]
  32. M.-J. Ni, R. Munipalli, P. Huang, N.B. Morley and M.A. Abdou, 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 (2007) 205–228. [CrossRef] [MathSciNet] [Google Scholar]
  33. M.-J. Ni, R. Munipalli, N.B. Morley, P. Huang and M.A. Abdou, 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 (2007) 174–204. [CrossRef] [MathSciNet] [Google Scholar]
  34. A. Prohl, Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. M2AN Math. Model. Numer. Anal. 42 (2008) 1065–1087. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  35. W. Qiu and K. Shi, Analysis of a semi-implicit structure-preserving finite element method for the nonstationary incompressible magnetohydrodynamics equations. Comput. Math. Appl. 80 (2020) 2150–2161. [CrossRef] [MathSciNet] [Google Scholar]
  36. D. Schötzau, Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96 (2004) 771–800. [Google Scholar]
  37. G. Tóth, The ∇ ・ B = 0 constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161 (2000) 605–652. [CrossRef] [MathSciNet] [Google Scholar]
  38. X. Zhang and Q. Ding, Coupled iterative analysis for stationary inductionless magnetohydrodynamic system based on charge-conservative finite element method. J. Sci. Comput. 88 (2021) 1–32. [CrossRef] [MathSciNet] [Google Scholar]
  39. X. Zhang and Q. Ding, A decoupled, unconditionally energy stable and charge-conservative finite element method for inductionless magnetohydrodynamic equations. Comput. Math. Appl. 127 (2022) 80–96. [CrossRef] [MathSciNet] [Google Scholar]
  40. X. Zhang and X. Wang, A fully divergence-free finite element scheme for stationary inductionless magnetohydrodynamic equations. J. Sci. Comput. 90 (2022) 70. [CrossRef] [Google Scholar]
  41. G. Zhang, Y. He and D. Yang, Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain. Comput. Math. Appl. 68 (2014) 770–788. [CrossRef] [MathSciNet] [Google Scholar]
  42. G. Zhang, J. Yang and C. Bi, Second order unconditionally convergent and energy stable linearized scheme for MHD equations. Adv. Comput. Math. 44 (2018) 505–540. [CrossRef] [MathSciNet] [Google Scholar]
  43. J. Zhao, Analysis of finite element approximation for time-dependent Maxwell problems. Math. Comput. 73 (2004) 1089–1105. [Google Scholar]

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