High-order Temporal Discrete Finite Element Methods For Two Classes Of Coupled Models Of Maxwell's Equations

Maxwell's equations are the pivotal part of the theory of electrical and magnetic fields,and have been widely applied into quantum mechanics,optical fiber communication,radar,medical and so on.Based on Maxwell's equations,two new equation groups,the linear nonlocal Drude model and the incompressible magnetohydrodynamic(MHD)equations,could be derived.In this thesis,numerical schemes with high-order temporally accuracy and computational efficiency have been designed and analyzed for these equations.This thesis briefly introduces the linear nonlocal Drude model as well as the discontinuous Galerkin(DG)methods.We propose a decoupled second-order backward differentiation formula(BDF)DG methods for the Drude model.In detail,the second-order BDF is used in time discretization,combined with the DG methods in spatial discretization,and consequently a corresponding decoupled BDF-DG numerical scheme has been constructed.The energy stability of the spatial semi-discrete scheme and the fully discrete scheme is theoretically proved.In the fully discrete system,to improve the computational efficiency,we employ a "decoupling" technique,which divides the whole system into two smaller ones.More precisely,the current density is explicit in the electric field equations,while the electric field is explicit in the current density equations.This "decoupled"technique would bring more advantages in computational efficiency when the system is relatively larger.In addition,the analysis of the optimal convergence rates of the fully discrete scheme is given.Finally,the numerical experiments are carried out to verify the theoretical analysis.The numerical results show that the decoupled BDF-DG scheme possesses the properties of second-order temporal and spatial accuracy and energy decaying,which is consistent with the theoretical analysis.For the incompressible MHD equations,by utilizing the "zero-energy-contribution"property,we design a fully decoupled linear numerical scheme with the second-order BDF finite element methods(FEM).Meanwhile,the detailed implementation of the numerical scheme will be described.Precisely,in the scheme,the pressure field appears explicitly in the velocity equation,and would be computed in the form of Poisson equation,named as"decoupled",which could remove the incompressible constraint from the fluid equation.Through treating the coupling terms explicitly,the magnetic and velocity field equations are reduced to elliptic equations with constant coefficients.This results in a symmetric positive definite matrix in practical computation,which can therefore be solved very efficiently by using the conjugate gradient method.It is relatively fast to compute the "zeroenergy-contribution" function we define by direct algebraic calculation and the pressure field by solving the Poisson equation.As a result,this scheme is more friendly to computation than traditional decoupled schemes.Moreover,smaller subsystems allow us to adopt finer meshes,which in turn can lead to higher accuracy.Theoretically,the unconditional energy stability,unique solvability and optimal error estimates of the system are rigorously proved.Finally,several numerical examples are used to verify these properties as well as the efficiency.