Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 6, p. 1065-1087

The incompressible MHD equations couple Navier-Stokes equations with Maxwell’s equations to describe the flow of a viscous, incompressible, and electrically conducting fluid in a Lipschitz domain $\Omega \subset {ℝ}^{3}$. We verify convergence of iterates of different coupling and decoupling fully discrete schemes towards weak solutions for vanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouples the computation of velocity field, pressure, and magnetic fields at every iteration step.

DOI : https://doi.org/10.1051/m2an:2008034
Classification:  65N30
Keywords: magneto-hydrodynamics, discretization, FEM, fixed-point scheme, splitting-method
@article{M2AN_2008__42_6_1065_0,
author = {Prohl, Andreas},
title = {Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {42},
number = {6},
year = {2008},
pages = {1065-1087},
doi = {10.1051/m2an:2008034},
zbl = {1149.76029},
mrnumber = {2473320},
language = {en},
url = {http://www.numdam.org/item/M2AN_2008__42_6_1065_0}
}

Prohl, Andreas. Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 6, pp. 1065-1087. doi : 10.1051/m2an:2008034. http://www.numdam.org/item/M2AN_2008__42_6_1065_0/

[1] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823-864. | MR 1626990 | Zbl 0914.35094

[2] F. Armero and J.C. Simo, Long-time dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations. Comp. Meth. Appl. Mech. Engrg. 131 (1996) 41-90. | MR 1393572 | Zbl 0888.76042

[3] L. Banas and A. Prohl, Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations. (In preparation).

[4] D. Boffi, P. Fernandes, L. Gastaldi and I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 (1999) 1264-1290. | MR 1701792 | Zbl 1025.78014

[5] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer (1994). | MR 1278258 | Zbl 0804.65101

[6] L. Cattabriga, Su un problema al contorno relativo al sistemo di equazioni di Stokes. Rend. Sem Mat. Univ. Padova 31 (1961) 308-340. | Numdam | MR 138894 | Zbl 0116.18002

[7] Z. Chen, Q. Du and J. Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37 (2000) 1542-1570. | MR 1759906 | Zbl 0964.78017

[8] A.J. Chorin, Numercial solution of the Navier-Stokes equations. Math. Comp. 22 (1968) 745-762. | MR 242392 | Zbl 0198.50103

[9] M. Costabel and M. Dauge, Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93 (2002) 239-277. | MR 1941397 | Zbl 1019.78009

[10] V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds. Ann. Math. Pura Appl. 122 (1979) 159-198. | MR 565068 | Zbl 0432.58026

[11] J.-F. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87 (2000) 83-111. | MR 1800155 | Zbl 0988.76050

[12] J.-F. Gerbeau, C. Le Bris and T. Lelievre, Mathematical methods for the magnetohydrodynamics of liquid crystals. Oxford Science Publication (2006). | Zbl 1107.76001

[13] V. Girault, R.H. Nochetto and R. Scott, Maximum-norm stability of the finite element Stokes projection. J. Math. Pures Appl. 84 (2005) 279-330. | MR 2121575 | Zbl pre02164965

[14] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Springer (1986). | MR 851383 | Zbl 0585.65077

[15] M.D. Gunzburger, A.J. Meir and J.S. Peterson, On the existence and uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comp. 56 (1991) 523-563. | MR 1066834 | Zbl 0731.76094

[16] U. Hasler, A. Schneebeli and D. Schötzau, Mixed finite element approximation of incompressible MHD problems based on weighted regularization. Appl. Numer. Math. 51 (2004) 19-45. | MR 2083323 | Zbl 1126.76341

[17] J.G. Heywood and R. Rannacher, Finite element solution of the nonstationary Navier-Stokes problem, I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275-311. | MR 650052 | Zbl 0487.76035

[18] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237-339. | MR 2009375 | Zbl 1123.78320

[19] T.J.R. Hughes, L.P. Franca and M. Balestra, A new finite element formulation for computational fluid mechanics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolation. Comp. Meth. Appl. Mech. Eng. 59 (1986) 85-99. | MR 868143 | Zbl 0622.76077

[20] F. Kikuchi, On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sec. IA 36 (1989) 479-490. | MR 1039483 | Zbl 0698.65067

[21] P. Monk, Finite element methods for Maxwell's equations. Oxford University Press, New York (2003). | MR 2059447 | Zbl 1024.78009

[22] A. Prohl, Projection and quasi-compressibility methods for solving the incompressible Navier-Stokes equations. Teubner-Verlag, Stuttgart (1997). | MR 1472237 | Zbl 0874.76002

[23] A. Prohl, On the pollution effect of quasi-compressibility methods in magneto-hydrodynamics and reactive flows. Math. Meth. Appl. Sci. 22 (1999) 1555-1584. | MR 1722949 | Zbl 0937.76070

[24] A. Prohl, On pressure approximation via projection methods for nonstationary incompressible Navier-Stokes equations. SIAM J. Numer. Anal. (to appear). | MR 2452856

[25] D. Schötzau, Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96 (2004) 771-800. | MR 2036365 | Zbl 1098.76043

[26] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36 (1983) 635-664. | MR 716200 | Zbl 0524.76099

[27] R. Temam, Sur l'approximation de la solutoin des equations de Navier-Stokes par la méthode de pas fractionnaires II. Arch. Rat. Mech. Anal. 33 (1969) 377-385. | MR 244654 | Zbl 0207.16904

[28] J. Zhao, Analysis of finite element approximation for time-dependent Maxwell problems. Math. Comp. 73 (2003) 1089-1105. | MR 2047079 | Zbl 1119.65392