The Finite Element Methods For Unsteady Incompressible Magnetohydrodynamics Equations

This paper is devoted to studying two kinds of finite element methods for the di-mensionless incompressible magnetohydrodynamic(MHD)equations.First,we consider the non-stationary incompressible MHD equations as follows where u represents the velocity of the fluid flow,B represents magnetic field,p represents the pressure,f and g represent the external body force terms.The MHD equations are characterized by three parameters:Re the fluid Reynolds number,? the magnetie permeability and ? the electric conductivity.Second,two or three dimensional unsteady incompressible MHD equations cotain-ing an electric field term are considered as follows with the following initial boundary conditions where the coefficients are the fluid Reynolds number Re which satisfies Re-1= v,the coupling number S and the magnetic Reynolds number Rm.j = E +u x B represents the Ohm's law.The term j x B is the Lorentz force,a consequence of the electric current j running within the magnetic field B.It is a force that influences the motion along the velocity field u,of the particles of the conducting fluid.The term f is due to possible external force.A typical case for such forces is that of the gravity force,where n is the unit outer normal vector.For the sake of simplicity,we assume that all the parameters are positiveMagnetohydrodynamics is the study of the interaction of electromagnetic fields and conducting fluids.The modeling consists of a coupling between the equations of Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism via Lorentz's force and Ohm's Law.It arises in many applications not only in many branches of physics but also engineering technology,such as wave propagation in iono-sphere in geophysics:MHD engine:control of MHD boundary layer and revival of liquid-metal MHD electricity generation.Therefore,it is necessary to study the algorithm of magnetohydrodynamic equations.In the third chapter,a fully discrete defect correction finite element method for the unsteady incompressible MHD equations,which is leaded by combining the Back Euler time discretization with the two-step defect correction in space,is presented.The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement Firstly,the nonlinear MHD equation is solved with an artificial viscosity term.Then.the numerical solutions are improved on the same grid by a linearized defect-correction technique.Then,we introduce the numerical analysis including stability analysis and error analysis.The numerical analysis proves that our method is stable and has an optimal convergence rateIn the fourth chapter,a time discrete projection method is given.The key of the method is that it naturally preserves the important Gauss's law,namely(?)·B =:.In contrast to most previous approaches that eliminate the electrical field variable E and give a direct discretization of the magnetic field.Our new method preserves the electrical field variable E and uses the relation of the fluid velocity variable u,the magnetic field variable B and the electrical field variable E.It is shown that the algorithm allows for a discrete energy inequality and is unconditional stable.Then the optimal error analysis is given.Finally,we present some numerical results to confirm our analysis and show clearly the stability and optimal convergence of the projection finite element method for the time-dependent MHD equations.