Optimal error estimates of a decoupled, linear, unconditionally stable and charge-conservative finite element scheme for a thermally coupled incompressible IMHD system

We develop and analyze a decoupled, linear, unconditionally stable, charge-conservative fully discrete scheme for a thermally coupled incompressible inductionless magneto-hydrodynamics (IMHD) system. Due to the nonlinearity and coupling of the system, developing a decoupled and practically effective scheme has always been a challenging problem. We overcome the difficulty by discretizing the temporal variables with the backward Euler method, linearizing the nonlinear convection term and decoupling the velocity, current density and temperature with implicit-explicit (IMEX) techniques, as well as decoupling the velocity and pressure with pressure-projection method. By discretizing the spatial variables with the finite element method, we acquire high accuracy. It is worth noting that the scheme is easy to implement since it requires solving merely a linear subsystem at each time step. Moreover, it is unconditionally stable, and yields an exactly divergence free current density directly. By introducing various finite element projections and developing novel inductive reasoning techniques, we gain optimal $L^2$ error estimates for the velocity, temperature, current density, pressure and electric potential. Numerical tests are provided to verify the good performance of the scheme such as the accuracy and charge conservation.