Conservative nonconforming virtual element method for stationary incompressible magnetohydrodynamics

Abstract: In this paper, we propose a conservative nonconforming virtual element method for the full stationary incompressible magnetohydrodynamics model. We leverage the virtual element satisfactory divergence-free property to ensure mass conservation for the velocity field. The condition of the well-posedness of the proposed method, as well as the stability are derived. We establish optimal error estimates in the discrete energy norm for both the velocity and magnetic field. Furthermore, by employing a new technique, we obtain the optimal error estimates in $L^2$-norm without any additional conditions. Finally, numerical experiments are presented to validate the theoretical analysis. In the implementation process, we adopt the effective Oseen iteration to handle the nonlinear system.