Second Order Unconditionally Convergent Fully Discrete Scheme for Incompressible Vector Potential MHD System

In this paper, we present and analyze a second order unconditionally convergent mixed finite element method for solving incompressible magnetohydrodynamic problem based on magnetic vector potential. We utilize the second-order backward difference formula within the framework of the mixed finite element method and linearize the nonlinear terms by using the extrapolated technique. The finite element approximation is proposed with the fluid equations are discretized by Taylor–Hood type elements and the magnetic vector potential equation by edge elements. As a result, the divergence-free condition on the magnetic induction preserves exactly on the discrete level. Unconditional error estimates for the velocity and magnetic vector potential are established in the sense that no any restrictions are imposed on the relationship between the time-step size and the spatial size. Finally, several three-dimensional numerical simulations are carried out to illustrate the developed scheme, including convergence tests and some benchmark problems, such as the driven cavity problems and hydromagnetic Kelvin–Helmholtz instability