This article presents an unconditionally stable mixed finite element method for time-dependent inductionless magnetohydrodynamics (MHD) problem. We first propose a mixed variational formulation based on the variables ( u, p, j, ϕ). A fully discrete second-order Crank-Nicolson extrapolation scheme based on mixed finite element method is considered, in which the Navier-Stokes equations are approximated by MINI-elements and the current density is discretized by the divergence-conforming Raviart-Thomas element. A notable feature of this method is that the discrete density maintains charge conservation property. It is shown that the scheme is unconditionally stable. We prove that the optimal estimates on the low regularity hypothesis for the exact solutions. Finally, some numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

In this paper, we consider the nonstationary magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation. The second order backward difference formula is used for time derivative terms, and the mixed finite method is used for spatial discretization, we employ the Taylor-Hood elements to approximate heat and Navier-Stokes equations, N$\mathrm{\acute{e}}$d$\mathrm{\acute{e}}$lec edge elements are used to approximate the magnetic induction. The divergence free conditions are weakly satisfied at the discrete level. Due to the use of N$\mathrm{\acute{e}}$d$\mathrm{\acute{e}}$lec edge element, the proposed method is particularly suitable for problems defined on non-smooth and multi-connected domains. Moreover, the numerical scheme is energy conserving. Under the weak regularity hypothesis of the exact solution, we present error estimate for velocity, magnetic variable and temperature. Finally, the convergence analysis is verified by some experiments, and the magnetic fluid phenomenon is simulated by driven cavity flow.