Structure-preserving Finite Element Methods For Magnetohydrodynamic Equations

Magnetohydrodynamics is proposed by Swedish physicist Hannes Alfven.It is a subject to describe the motion of unmagnetized conductive fluid under the action of elec-tromagnetic field.It is widely used in astrophysics,controlled thermonuclear reaction,aerospace engineering and other research fields.The basic equations of magnetohydrody-namics are composed of Navier-Stokes equations in hydrodynamics and Maxwell equa-tions in electromagnetics through Lorentz force.Magnetohydrodynamics equations are nonlinear partial differential equations.In general,the classical solutions are difficult to obtain and can only be solved numerically.Based on the above background,in this paper,we will systematically study the structure-preserving finite element methods for magne-tohydrodynamics equations.We will first study Maxwell equation with mixed finite ele-ment methods,and then study magnetohydrodynamics equation with structure-preserving mixed finite element methods.For Maxwell equation,the corresponding energy estimates and error analyses are given for weak scheme,semi-discrete scheme and full-discrete scheme respectively.The Crank-Nicolson scheme is used to discretize the time variables.Based on the existing research,by selecting the more appropriate operators,the regularity requirement of the classical solution in the error estimation formula is reduced,and the theoretical results are optimized.Through numerical experiments,it is verified that the variables meet the requirements of error and convergence order,as well as the divergence-free condition of magnetic field.For the magnetohydrodynamics equation,by choosing appropriate finite element dis-crete space,the strict divergence-free conditions of velocity field and magnetic field are naturally satisfied,so the energy-preserving can be easily proved.The above two condi-tions are the innovation of this paper.The time variable is discretized by Crank-Nicolson scheme,and the nonlinear full-discrete scheme is obtained.The Newton iteration scheme of the full-discrete scheme is derived,which transforms the nonlinear problem into a linear one.Through numerical experiments,the law of energy dissipation and the divergence-free condition of velocity field and magnetic field are verified.