• DOI:10.1051/M2AN/2018006
• Corpus ID: 125702088

A priori estimates and optimal finite element approximation of the MHD flow in smooth domains

@article{He2018APE,
  title={A priori estimates and optimal finite element approximation of the MHD flow in smooth domains},
  author={Yinnian He and Jun Zou},
  journal={Mathematical Modelling and Numerical Analysis},
  year={2018},
  volume={52},
  pages={181-206}
}

• Yinnian He, J. Zou
• Published 2018
• Mathematics
• Mathematical Modelling and Numerical Analysis

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 ∈ L 4 (0,T ;L 2 (Ω)3 × 3 ) and ∇ × B ∈ L 4 (0,T ;L 2 (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 L 2 -norm, and the optimal error estimates of the discrete velocity and magnetic field in L 2 -norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.