Stationary Mhd Mixed Finite Element Method

A nonlinear Galerkin mixed element (NGME) method, A Galerkin/Petrov- least squares mixed finite element method, and A nonlinear Galerkin/Petrov- least squares mixed finite element (NGPLSME) method for the stationary incompressible magneto-hydrodynamics equations are derived.This paper consists of in the following three parts.Part I. A nonlinear Galerkin mixed element (NGME) method is presented. And the existence and error estimates of the NGME solution are discussed.Part II. The stationary incompressible magnetohydrodynamics problems are dealt with Galerkin/Petrov- least squares mixed finite element method. A Galerkin/Petrov-least squares mixed finite element formulation for the stationary incompressible magnetohydrodynamics problems is presented. The existence, uniqueness and convergence of the discrete solution is proved.Part III. A NGPLSME method for the stationary incompressible magnetohydrodynamics problems is presented. The existence and uniqueness of the NGPLSME solution are discussed. And convergence(at optimal rate) of the NGPLSME solution is proved in case of sufficient viscosity (or small data).