In this work, we proposed a new fully discrete linearized $H^1$-conforming Lagrange FEM for the two-dimensional megneto-hydrodynamics equations based on a magnetic potential formulation such that the numerical solutions would converge not only in convex and smooth domains but also in nonconvex and nonsmooth domains. The convergence of subsequences of the numerical solutions is proved only based on the regularity of the initial conditions and source terms, without extra assumptions on the regularity of the solution. Numerical examples are given to support the theoretical analysis.
