Optimal error analysis of an unconditionally stable BDF2 finite element approximation for the 3D incompressible MHD equations with variable density.

This paper presents a second-order finite element scheme for the approximations of the three-dimensional (3D) incompressible magnetohydrodynamics system with variable density (VD-MHD), where two-step backward differentiation formula (BDF2) is used to discrete the time derivative and the (P 2 , P 1 b , P 1 , P 1) finite elements are used to approximate the density, velocity, pressure and magnetic. Based on an equivalent VD-MHD system, the proposed finite element algorithm is unconditionally stable in the sense that the discrete energy inequalities hold without small condition of the time step size. After a rigorous analysis, the optimal L 2 error estimates (τ 2 + h 2) of the density, velocity field and magnetic field are established, where τ and h are the time step size and mesh size. Finally, numerical results are given to support these convergence rates. [ABSTRACT FROM AUTHOR]