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 
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     title = {A priori estimates and optimal finite element approximation of the {MHD} flow in smooth domains},
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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/

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