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

Abstract

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 (P2,P1b,P1,P1) 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 L2 error estimates (τ2+h2) 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.