  A new fully discrete linearized $H^1$-conforming Lagrange finite element
method is proposed for solving the two-dimensional magneto-hydrodynamics
equations based on a magnetic potential formulation. The proposed method yields
numerical solutions that converge in general domains that may be nonconvex,
nonsmooth and multi-connected. 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.
Strong convergence in $L^2(0,T;{\bf L}^2(\Omega))$ was proved for the numerical
solutions of both ${\bm u}$ and ${\bm H}$ without any mesh restriction.
