The Oseen Type Finite Element Iterative Method for the Stationary Incompressible Magnetohydrodynamics (Advances in Applied Mathematics and Mechanics, 2018)
In this article, by applying the Stokes projection and an orthogonal projectionwith respect to curl and div operators, some new error estimates of finite elementmethod (FEM) for the stationary incompressible magnetohydrodynamics (MHD) areobtained. To our knowledge, it is the first time to establish the error bounds which areexplicitly dependent on the Reynolds numbers, coupling number and mesh size. Onthe other hand, The uniform stability and convergence of an Oseen type finite elementiterative 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 ofequations) are analyzed under the condition that $h$≤(‖F‖$_{−1}$/‖F‖$_0$)$^{1/2}δ$. Finally, somenumerical tests are presented to demonstrate the effectiveness of this algorithm.