Unconditional Stability Of The Modified Characteristics FEMS For Fluid Dynamics Equations

This dissertation is mainly devoted to studying the unconditional stability of the modified characteristic finite element method for equations of fluid dynamics.The mag-netohydrodynamic(MHD)equations are the study of the interaction of electromagnetic fields and conducting fluids.The model consists of a coupling between the equations of continuum fluid mechanics and the Maxwell's equation of electromagnetism via Lorentz's force and Ohm's law.we consider the incompressible MHD equations as follows with the following initial boundary conditionswhere u represents the velocity of the fluid,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 magnetic permeability and ? the electric conductivity.The model of rate type such as viscoelastic Oldroyd fluids can predict the stress relaxation as well as the retardation of deformation and become popular for describing polymeric suspension.The initial boundary value problem of unsteady viscoelastic Oldroyd flow is considered:where ?(t)=pexp(-?t),1/? is the relaxation time,??0 is the viscoelastic coefficient number.Here,u represents the velocity vector,p is the pressure,f is the body force.v=1/Re is the viscosity numberIn the third chapter,the characteristics finite element method is applied to MHD equations.Firstly,a characteristic time-discrete system is presented,which can get the corresponding stability and error estimate.Then,the modified characteristics finite element method for the time-dependent MHD equations problem is presented,which can get the corresponding stability and error estimates.In this chapter we present the numerical analysis of smooth solution and Hartmann flow.The results show that our method has unconditional stability and optimal error estimates.In the fourth chapter,the characteristics finite element method is applied to Oldroyd fluids.Firstly,a char-acteristic time-discrete system is presented,which can get the corresponding stability and error estimate.Then,the modified characteristics finite element method for the time-dependent Oldroyd fluids problem is presented,which can get the corresponding stability and error estimates.In this chapter we present the numerical analysis including the stability and error estimates.The results show that our method has unconditional stability and optimal error estimates.