Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain

This paper considers finite element approximation and three coupled type iterations of stationary magnetohydrodynamics (MHD) equations on a general Lipschitz domain. An additional Lagrange multiplier rr is introduced related to divergence free of magnetic field b and the b is analyzed in H(curl;Ω) space, which is proposed in Schötzau (2004). In finite element discretization, the hydrodynamic unknowns are approximated by stable finite element pairs, and the magnetic unknown is discretized by curl-conforming Nédélec element spaces. The well-posedness of this formula and the optimal error estimate are provided. Based on this, for numerical implementation of this scheme, we propose and discuss three coupled type iterative methods which are stable and convergent under different conditions. Specifically, Iteration I is stable and convergent under strong condition. Iteration II is stable and convergent under weaker condition. Iteration III is unconditionally stable and convergent under the weakest condition. Finally, some numerical tests confirm our theoretical analysis.