Abstract
This paper is devoted to the study of finite element method for the isentropic compressible magnetohydrodynamics system. We employ quadratic finite elements to approximate the velocity and Nedelec edge elements to approximate the magnetic induction. The continuity equation is approximated by Discontinuous Galerkin method. Based on the renormalized scheme, we derive the stability of the proposed numerical scheme for compressible magnetohydrodynamics equations. With the help of the theory of the topological degree, the existence of solution to the numerical scheme is proved. Some techniques have to be adopted to improve the integrability of density so as to achieve strong convergence of the discrete density. As both meshwidth and timestep size tend to zero, we show that finite element solution converges to a global weak solution of the continuous problem. The results of this paper can be regarded as a numerical version of the existence analysis of the compressible MHD system.
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