A Low Order Globally Divergence-Free WG Finite Element Method for Steady Thermally Coupled Incompressible MHD Flow

This paper constructs a low-order weak Galerkin (WG) finite element method for steady thermally coupled incompressible magnetohydrodynamic flows. Inside each element, piecewise linear polynomials are adopted to approximate velocity, magnetic field and temperature, while piecewise constant functions are used for pressure and magnetic pseudo-pressure. On element interfaces, piecewise constants are employed for the numerical traces of velocity and temperature, and piecewise linear polynomials are utilized for those of magnetic field, pressure and magnetic pseudo-pressure. The proposed WG method can produce globally divergence-free numerical approximations for both velocity and magnetic field fields. We establish the existence and uniqueness of discrete solutions, and derive optimal a priori error estimates for the fully discrete scheme. Meanwhile, a convergent linearized iterative algorithm is designed for numerical implementation. Several numerical experiments are carried out to verify the correctness of the theoretical analysis.