  In this paper, we consider the numerical approximation for a diffuse
interface model of the two-phase incompressible inductionless
magnetohydrodynamics problem. This model consists of Cahn-Hilliard equations,
Navier-Stokes equations and Poisson equation. We propose a linear and decoupled
finite element method to solve this highly nonlinear and multi-physics system.
For the time variable, the discretization is a combination of first-order Euler
semi-implicit scheme, several first-order stabilization terms and
implicit-explicit treatments for coupling terms. For the space variables, we
adopt the finite element discretization, especially, we approximate the current
density and electric potential by inf-sup stable face-volume mixed finite
element pairs. With these techniques, the scheme only involves a sequence of
decoupled linear equations to solve at each time step. We show that the scheme
is provably mass-conservative, charge-conservative and unconditionally energy
stable. Numerical experiments are performed to illustrate the features,
accuracy and efficiency of the proposed scheme.
