The Arrow–Hurwicz iterative finite element method for the stationary magnetohydrodynamics flow

Abstract

In this paper, we propose and analyze an Arrow–Hurwicz iterative finite element method for solving the stationary incompressible magnetohydrodynamics (MHD) equations discretized by mixed element methods. Comparing with the other iterative methods, the significant feature of this iterative method is that it does not need to solve any saddle-point system at each iteration step except for determination of the initial functions. Under several reasonable conditions, it is proved that the solution solved by the proposed iterative method is uniformly bounded with respect to the mesh width h and the iteration number n and the method converges geometrically with a contraction number independent of the finite element mesh width h. Ample numerical experiments are performed to validate the accuracy and the efficiency of the numerical scheme.