Stability and convergence of two-level iterative methods for the stationary incompressible magnetohydrodynamics with different Reynolds numbers

Abstract In the paper we develop some two-level finite element iterative methods and use these methods to solve the stationary incompressible magnetohydrodynamics (MHD) with different Reynolds numbers under some different uniqueness conditions. Firstly, we use the Stokes type iterative method, Newton type iterative method and Oseen type iterative method of m times on a coarse mesh with mesh size H and then solve a linear problem with the Stokes type, Newton type and Oseen type correction of one time on a fine grid with mesh sizes h ≪ H . Furthermore, we analyze the uniform stability and convergence of these methods with respect to Reynolds numbers, mesh sizes h and H and iterative times m. Finally, the stationary incompressible MHD equations with large Reynolds number are researched by the one-level finite element method based on the Oseen type iteration on a fine mesh under a weak uniqueness condition.

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