An Arrow-Hurwicz iterative method based on charge-conservation for the stationary inductionless magnetohydrodynamic system

In this paper, an Arrow-Hurwicz iterative finite element method based oncharge-conservation for solving the stationary inductionless magnetohydrodynamic equations is proposed and analyzed. The current density and electric potential are discretized by \mathbf{H}_0(\mathrm{div},\Omega)\times L^2(\Omega)-conforming finite element pairs,respectively, which maintains charge-conservation. The hydrodynamic unknowns are discretized by stable velocity-pressure finite element pairs. Furthermore, we use the Arrow-Hurwicz iterative method to decouple the velocity and pressure, which leads to that there are no any large-scale saddle-point system need to be solved at each iteration step except for determination of the initial functions. It has been proven that the proposed method converges geometrically with a contraction number independent of the mesh width. Numerical results are presented to verify the results of theoretical analysis and effectiveness of the proposed method. Mathematics Subject Classification (2000) 65N15 · 65N30 · 65N12 · 65M12