Optimal error estimates of two-level iterative finite element methods for the thermally coupled incompressible MHD with different viscosities

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 H$$ H $$ 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 hMUCH LESS-THANH$$ h\ll H $$ 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 h,H$$ h,H $$ and iterative times m$$ m $$. Extensive numerical results are presented to demonstrate the established theoretical findings and show the performances of these two-level iterative schemes with different viscosities.