(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
In this article, by applying the Stokes projection and an orthogonal projection with respect to curl and div operators, some new error estimates of finite element method (FEM) for the stationary incompressible magnetohydrodynamics (MHD) are obtained. To our knowledge, it is the first time to establish the error bounds which are explicitly dependent on the Reynolds numbers, coupling number and mesh size. On the other hand, The uniform stability and convergence of an Oseen type finite element iterative method for MHD with respect to high hydrodynamic Reynolds number R_e and magnetic Reynolds number R_m, or small δ=1−σ with (C_0, C_1 are constants depending only on Ω and F is related to the source terms of equations) are analyzed under the condition that h≤(‖F‖_−1/‖F‖_0)^1/2δ. Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.