A PRIORI ESTIMATES AND OPTIMAL FINITE ELEMENT APPROXIMATION OF THE MHD FLOW IN SMOOTH DOMAINS

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 del u is an element of L-4 (0, T; L-2 (Omega)(3x3)) and del x B is an element of L-4(0, T; L-2(Omega)(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.