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A new fully discrete linearized $H^1$-conforming Lagrange finite elementmethod is proposed for solving the two-dimensional magneto-hydrodynamicsequations based on a magnetic potential formulation. The proposed method yieldsnumerical solutions that converge in general domains that may be nonconvex,nonsmooth and multi-connected. The convergence of subsequences of the numericalsolutions is proved only based on the regularity of the initial conditions andsource 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 numericalsolutions of both ${\bm u}$ and ${\bm H}$ without any mesh restriction.

A convergent linearized Lagrange finite element method for the magneto-hydrodynamic equations in 2D nonsmooth and nonconvex domains