Error estimates of the fully discrete finite element method for thermally driven MHD problems including Joule heating

Abstract The main aim of this paper is to study the error estimate of fully discrete finite-element approximation for the time-dependent magnetohydrodynamic system coupled thermally equation through the well-known Boussinesq approximation, in which the Joule effect and Viscous heating are taken into account. To this end, we first prove the uniqueness of the weak solution to the model problem with a slightly smoother regularity assumption. The discretized problem is proposed by the Euler semi-implicit method for temporal discretization and by mixed finite elements for spatial discretization. Next, the well-posedness of the solution to the Euler semi-implicit semi-discrete scheme and the corresponding temporal semi-discrete error are established. Then, we consider the fully discrete Euler semi-implicit scheme based on the mixed finite method to approximate the fluid equation and Nédélec edge elements to approximate the magnetic induction. This fully discrete scheme requires only solving a linear system per time step. With a proper regularity assumption on the exact solution, optimal $L^{2}$-error estimates for the velocity, magnetic induction and temperature are derived unconditionally. Finally, several numerical examples are performed to demonstrate both the accuracy and efficiency of our proposed scheme.