Fully Discrete Finite Element Approximation of the MHD Flow

Abstract

Abstract In this work, we consider a fully discrete finite element approximation of the 3D incompressible magnetohydrodynamic system. The velocity and magnetic field are approximated both by piecewise quadratic finite elements, while the pressure is approximated by piecewise linear finite elements. The time discretization is based on the Crank–Nicolson scheme for the linear terms in the model and the explicit Adams–Bashforth for the nonlinear terms. We establish the optimal error estimates of both the approximate velocity and magnetic field in H 1 \mathbf{H}^{1} -norm and of the approximate pressure in L 2 L^{2} -norm. In order to achieve the optimal L 2 L^{2} -norm error estimates of both the approximate velocity and magnetic field, we shall make use of a special negative norm technique.