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.
Mots-clés : MHD flow, finite element approximations, a priori estimates, error estimates, negative-norm technique
@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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/m2an/2018006/
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