| This paper mainly studies the pressure-correction finite element methods for the first and second order schemes of the incompressible magnetohydrodynamic(MHD)equations.Two or three dimensional unsteady incompressible MHD equations are con-sidered as follows#12 with the following initial boundary conditions#12 where QT=?×(0,T)with ?(?)R2 and fixed T ?(0,?).Here,u,p,E,B are the velocity field,the pressure,the electric field and the magnetic field.In addition,f is the know source term.The MHD equations are characterized by two parameters:v the kinematic viscosity and statisfies v=Re-1.Re the fluid Reynolds number,S the coupling number and Rm the magnetic Reynolds number.where n is the unit outer normal.For the sake of simplicity,we assume all the parameters are positive.In the third chapter,we give a pressure-correction projection scheme in rotational form for the unsteady incompressible MHD equations:first-order schemes.This method uses the relations of the fluid velocity variable u.the magnetic field variable B and the electrical field variable E,the electrical field variable E were preserved.The theory analysis proves that the rotation form of the algorithm provides optimal error estimates in terms of the H1-norm of the velocity and of the L2-norm of the pressure.In the fourth chapter,we introduce the efficient numerical scheme for the unsteady incompressible MHD equations:second-order schemes.This algorithm applies the aux-iliary linear problem and divides the error estimation into two parts.One is related to time-dependent linear Stokes operator,and the other is related to nonlinear terms.Obviously,the main error term is obtained by the approximation of the linear operator,and the approximation error of the nonlinear term is relatively small,which is conve-nient to deal with.The theory analysis proves that the algorithm provides optimal error estimates in terms of the H1-norm of the velocity and of the L2-norm of the pressure. |
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