Unconditional stability and optimal error estimates of first order semi-implicit stabilized finite element method for two phase magnetohydrodynamic diffuse interface model

Highlights：

• This paper considers the first order semi-implicit energy stable finite element method for the two phase magnetohydrodynamic flows (the Cahn–Hilliard equation coupling the incompressible MHD system), some numerical results are provided to show the performances of the considered numerical scheme. The main contribution of this work is to present the rigorous unconditional stability of numerical solutions in both time semi-discrete and fully discrete cases with two suitable stabilized terms. Optimal error estimates of numerical solutions are also presented by using the energy method and the Gronwall lemma. Our findings enrich and develop the theoretical analysis results of mixed finite element for the two-phase incompressible flows.