In this paper, we develop and design the two‐level finite element iterative methods for the stationary thermally coupled incompressible MHD equations. The considered numerical schemes are based on the classical iterations with one correction. First, under some strong uniqueness conditions, the rough solutions are obtained on the coarse mesh with the mesh size and the Simple, Oseen, and Newton iterations, respectively. Three kinds of corrections are made by solving a linear problem on the fine mesh with mesh size with different viscosities. Finally, under the weak uniqueness condition, the stationary thermally coupled incompressible MHD is solved by the one‐level finite element Oseen iteration on the fine mesh. The uniform stability and convergence of these two‐level iterative methods are analyzed with respect to the mesh sizes and iterative times . Extensive numerical results are presented to demonstrate the established theoretical findings and show the performances of these two‐level iterative schemes with different viscosities.

In this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in and ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.