Volume 52, Number 1, January–February 2018
|Page(s)||181 - 206|
|Published online||04 May 2018|
A priori estimates and optimal finite element approximation of the MHD flow in smooth domains
School of Mathematics and Statistics, Xi’an Jiaotong University,
710049, P.R. China
2 Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong
* Corresponding author: e-mail: firstname.lastname@example.org
Accepted: 11 January 2018
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.
Mathematics Subject Classification: 65N30 / 35Q35 / 65N12 / 76M10 / 76W05
Key words: MHD flow / finite element approximations / a priori estimates / error estimates / negative-norm technique
© EDP Sciences, SMAI 2018
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