A Convergent Linearized Lagrange Finite Element Method for the Magneto-hydrodynamic Equations in Two-Dimensional Nonsmooth and Nonconvex Domains

Abstract

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 multiconnected. 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 $L^2(0,T;{L}^2(\Omega))$ was proved for the numerical solutions of both ${\bm u}$ and ${\bm H}$ without any mesh restriction.

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