A convergent linearized Lagrange finite element method for the magneto-hydrodynamic equations in 2D nonsmooth and nonconvex domains
It provides a provably convergent numerical method for MHD equations in general domains, addressing a gap in existing methods that require smooth domains or solution regularity.
The paper proposes a new linearized finite element method for 2D magneto-hydrodynamic equations that converges in nonsmooth, nonconvex domains without extra regularity assumptions, proving strong L2 convergence for both velocity and magnetic fields.
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 $L^2(0,T;{\bf L}^2(Ω))$ was proved for the numerical solutions of both ${\bm u}$ and ${\bm H}$ without any mesh restriction.