Convergence analysis of a conservative finite element scheme for the thermally coupled incompressible inductionless MHD problem

Abstract

In this paper, a stable mixed finite element method for unsteady inductionless magnetohydrodynamics (MHD) problem coupled heat equation is devised. We first propose a mixed variational formulation based on all the variables, in which the thermal equation and Navier-Stokes equations are approximated by mini finite element method and the current density is discretized by the divergence-conforming elements. The novel feature of this scheme is that the discrete current density keeps charge conservation property. It is shown that the fully discrete first order Euler semi-implicit scheme is well-posed and unconditionally stable. The existence and uniqueness of the weak solutions for continuous problem is established by a numerical version analysis. Furthermore, under the low regularity hypothesis for the exact solutions, we prove the optimal error estimates of the velocity, current density and electric potential. Finally, some numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.