An efficient two-grid algorithm based on Newton iteration for the stationary inductionless magnetohydrodynamic system

In this paper, we propose and analyze a two-grid algorithm based on Newton iteration for solving the stationary inductionless magnetohydrodynamic system. The method involves first solving a small nonlinear system on a coarse grid with grid size H, followed by solving two linear problems on a fine grid with grid size h. These linear problems share the same stiffness matrix but differ only in their right-hand sides. The scaling between the coarse and fine grids is improved by our new method, while the approximate solution retains the same order of convergence as that observed in conventional methods. Furthermore, $H_0(\mathrm{div},\Omega )\times L_0^2\left(\Omega \right)$-conforming finite element pairs are utilized to discretize the current density and electric potential, ensuring that the discrete current density is exactly divergence-free. Stability and convergence analyses are rigorously derived, and $L^2$-error estimates for the velocity are provided. Numerical experiments are presented to verify the theoretical predictions and demonstrate the efficiency of the proposed method.