Convergence Analysis of a BDF Finite Element Method for the Resistive Magnetohydrodynamic Equations

In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation (BDF) and the finite element method (FEM) to solve the incompressible resistive magnetohydrodynamic (MHD) equations. In the discrete scheme, the pressure variable in the fluid field equation is computed through a Poisson equation, and a linear and decoupled method is adopted to separate both the magnetic and the fluid field functions from the original system. As a result, the original system is divided into several sub-systems for which the numerical solutions can be obtained efficiently. We prove the unique solvability, the unconditional energy stability, and particularly optimal error estimates for the proposed scheme. Numerical results are presented to validate the theory of the scheme.