Error analysis of the linearized Crank-Nicolson FEM for the incompressible vector potential magnetohydrodynamic system

A linearized fully discrete Crank-Nicolson finite element scheme is proposed for solving the three-dimensional incompressible magnetohydrodynamic equations based on a magnetic vector potential formulation, where the magnetic induction is written to a rotation of magnetic vector potential. By using the MINI element and lowest order Nedelec edge element to approximate the velocity field and pressure of fluid and magnetic vector potential, respectively, the numerical solution of magnetic induction can preserve the exactly divergence-free condition in fully discrete level. Error estimates for the velocity field and magnetic vector potential are rigorously analyzed under some reasonable regularity assumptions of exact solution. Finally, numerical results are given to support the theoretical analysis.