The Oseen Type Finite Element Iterative Method for the Stationary Incompressible Magnetohydrodynamics

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 Re and magnetic Reynolds number R-m, or small delta = 1-sigma with sigma = root 2C(0)(2)max{1,root 2S(c)} parallel to F parallel to(-1)/(min{R-e(-1),ScC1Rm-1})(2) (C-0, C-1 are constants depending only on Omega and F is related to the source terms of equations) are analyzed under the condition that h <= (parallel to F parallel to(-1)/parallel to F parallel to(0))(1/2)delta. Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.