A priori estimates and optimal finite element approximation of the MHD flow in smooth domains
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 181-206.

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities and a priori estimates for the velocity, pressure and magnetic field (u, p, B) of the MHD system under the assumption that ∇u ∈ L4(0,T;L2(Ω)3 × 3) and ∇ × B ∈ L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure in L2-norm, and the optimal error estimates of the discrete velocity and magnetic field in L2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

DOI : 10.1051/m2an/2018006
Classification : 65N30, 35Q35, 65N12, 76M10, 76W05
Mots clés : MHD flow, finite element approximations, a priori estimates, error estimates, negative-norm technique
He, Yinnian 1 ; Zou, Jun 1

1
@article{M2AN_2018__52_1_181_0,
     author = {He, Yinnian and Zou, Jun},
     title = {A priori estimates and optimal finite element approximation of the {MHD} flow in smooth domains},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {181--206},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {1},
     year = {2018},
     doi = {10.1051/m2an/2018006},
     zbl = {1395.65143},
     mrnumber = {3808158},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2018006/}
}
TY  - JOUR
AU  - He, Yinnian
AU  - Zou, Jun
TI  - A priori estimates and optimal finite element approximation of the MHD flow in smooth domains
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 181
EP  - 206
VL  - 52
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2018006/
DO  - 10.1051/m2an/2018006
LA  - en
ID  - M2AN_2018__52_1_181_0
ER  - 
%0 Journal Article
%A He, Yinnian
%A Zou, Jun
%T A priori estimates and optimal finite element approximation of the MHD flow in smooth domains
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 181-206
%V 52
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2018006/
%R 10.1051/m2an/2018006
%G en
%F M2AN_2018__52_1_181_0
He, Yinnian; Zou, Jun. A priori estimates and optimal finite element approximation of the MHD flow in smooth domains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 181-206. doi : 10.1051/m2an/2018006. http://www.numdam.org/articles/10.1051/m2an/2018006/

[1] F. Armero and J.C. Simo, 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 (1996) 41–90. | DOI | MR | Zbl

[2] I. Babuska and A. Aziz, Survey lectures on the mathematical foundations of the finite element method with applications to partial differential equations, edited by A. Aziz. Academic Press, New York (1973). | MR

[3] L. Baňas and A. Prohl, Convergent finite element discretizations of the multi-fluid nonstationary incompressible Magnetohydrodynamics equations. Math. Comput. 79 (2010) 1957–1999. | DOI | MR | Zbl

[4] J.H. Bramble and J. Xu, Some estimates for a weighted L2-projection. Math. Comp. 56 (1991) 463–476. | MR | Zbl

[5] L. Cattabriga, Si un problem al contorno relativo al sistema di equazioni di Stokes. Rend. Semin. Mat. Univ. Padova 31 (1961) 308–340. | Numdam | MR | Zbl

[6] P.G. Ciarlet, Finite Element Method for Elliptic Equations. North-Holland Publishing, Amsterdam (1978). | MR

[7] J.-F. Gerbeau, A stabilized finite element method for the incompressible Magnetohydrodynamics equations. Numer. Math. 87 (2000) 83–111. | DOI | MR | Zbl

[8] J.-F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford Univerisity Press, Oxford (2006). | DOI | MR | Zbl

[9] V. Georgescu, Some boundary value problems for differenttial forms on compact Riemannian manifolds. Ann. Mat. Pura Appl. 4 (1979) 159–198. | DOI | MR | Zbl

[10] V. Girault and P.A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin, Heidelberg (1987). | MR | Zbl

[11] M.D. Gunzburger, A.J. Meir and J.S. Peterson, On the existence and uniqueness and the finite element approxiamtion of solutions of the equations of stationary incompressible Magneto-hydrodynamics. Math. Comp. 56 (1991) 523–563. | DOI | MR | Zbl

[12] M.D. Gunzburger, O.A. Ladyzhenskaya and J.S. Peterson, On the global unique solvability of initial-boundary value problems for the coupled modified Navier-Stokes Maxwell equations. J. Math. Fluid Mech. 6 (2004) 462–482. | DOI | MR | Zbl

[13] Y.N. He, Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J. Numer. Anal. 35 (2015) 767–801. | DOI | MR | Zbl

[14] J.G. Heywood and R. Rannacher, 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 (1982) 275–311. | DOI | MR | Zbl

[15] J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem III: Smooothing property and high order error estimates for spatial discretization. SIAM J. Numer. Anal. 25 (1988) 489–512. | DOI | MR | Zbl

[16] W.F. Hughes and F.J. Young, The Electromagneto-Hydrodynamics of Fluids. Wiley, New York (1966).

[17] J.D. Jackson, Classical Electrodynamics. Wiley, New York (1975). | MR | Zbl

[18] A. Kiselev and O. Ladyzhenskaya, On existence and uniqueness of solutions of nonstationary problem for viscous incompressible fluid. Izv.Akad. Nauk SSSR Ser. Math. 21 (1957) 655–680. | MR | Zbl

[19] O.A. Ladyzhenskaya and V. Solonnikov, Solution of Some Nonstationary Magnethydrodynamical Problems for Incompressible Fluid. Trusy Steklov Math. Inst. 69 (1960) 115–173. | MR

[20] J. Li, J. M. Melenk, B. Wohlmuth and J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl. Numer. Math. 60 (2010) 19–37. | DOI | MR | Zbl

[21] P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003). | DOI | MR | Zbl

[22] H.A. Navarro, L. Cabezas-Góvez, R.C. Silva, and A. N. Montagnoli, A generalized alternating-direction implicit scheme for incompressible magnetohydrodynamic viscous flows at low magnetic Reynolds number. Appl. Math. Comput. 189 (2007) 1601–1613. | DOI | MR | Zbl

[23] A. Prohl, Convergent finite element discretizations of the nonstationary incompressible Magnetohydrodynamics system. ESAIM: M2AN 42 (2008) 1065–1087. | DOI | Numdam | MR | Zbl

[24] N.B. Salah, A. Soulaimani, W.G. Habashi, A finite element method for magnetohydrodynamics. Comput. Methods. Appl. Mech. Eng. 190 (2001) 5867–5892. | DOI | MR | Zbl

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

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

[27] J.A. Shercliff, A Textbook of Magnetohydrodynamics. Pergmon Press, Oxford (1965). | MR

[28] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Spinger-Verlag, New York (1988). | DOI | MR | Zbl

[29] R. Temam, Navier-Stokes Equations, Theory and Numerical, 3rd edn. North-Holland, Amsterdam (1983). | MR | Zbl

[30] R. Temam, Induced trajectories and approxiamate inertial manifolds. ESAIM: M2AN 23 (1989) 541–561. | DOI | Numdam | MR | Zbl

[31] G. Yuksel, R. Ingram, Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers. Int. J. Numer. Anal. Model. 10 (2013) 74–98. | MR | Zbl

Cité par Sources :