Optimal convergent analysis of a linearized Euler finite element scheme for the 2D incompressible temperature-dependent MHD-Boussinesq equations

In this paper, we study a first-order Euler semi-implicit finite element scheme for the two-dimensional incompressible Boussinesq equations for magnetohydrodynamics convection with the temperature-dependent viscosity, electrical conductivity and thermal diffusivity. In finite element discretizations, the mini finite element is used to approximate the velocity and pressure, and the piecewise linear finite element is used to approximate the magnetic field and temperature. The unconditional stability of the proposed scheme is proved. By introducing three projection operators with variable coefficients and using the method of mathematical induction, we obtain optimal error estimates under a CFL type condition. Finally, numerical examples are provided to demonstrate these convergence rates.