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

## TL;DR

This paper introduces a new linearized finite element method for 2D magneto-hydrodynamics equations that converges reliably in complex, nonconvex domains without requiring high regularity assumptions.

## Contribution

It develops a fully discrete, linearized $H^1$-conforming Lagrange finite element method that guarantees convergence in general domains with minimal regularity assumptions.

## Key findings

- Proves convergence of numerical solutions in nonconvex, nonsmooth domains.
- Establishes strong convergence in $L^2$-norms for velocity and magnetic field.
- No mesh restrictions needed for convergence.

## 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 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.