Convergence analysis of finite element method for incompressible magnetohydrodynamics system with variable density

This paper rigorously analyzes a finite element method for incompressible magnetohydrodynamics flows with variable density. A fully discrete scheme based on the Euler semi-implicit method is proposed. The magnetic equation is approximated by N$\acute{e}$d$\acute{e}$lec edge elements, the density equation is approximated by Discontinuous Galerkin method, and the momentum equations are approximated by continuous elements. The numerical scheme is showed to satisfy the laws of mass conservation and energy conservation. In addition, we prove that the discrete density system satisfies the stability, consistency and convergence. Employing the Lax–Milgram theorem, the existence of solution to the fully discrete scheme is demonstrated. As both meshwidth and timestep size tend to zero, we prove that the fully discrete solution converges to a weak solution of the continuous problem.