Picard–Newton finite element iteration for the stationary incompressible magnetohydrodynamics equations

In this work, we propose a two-step finite element iteration for the stationary incompressible magnetohydrodynamics equations and provide a theoretical analysis. In finite element discretization, stable finite element pairs approximate the hydrodynamic unknowns, and continuous finite element is used to discretize the electromagnetic system. To efficiently solve the nonlinear discretized problem, our method involves initially applying the Picard iteration, followed by the Newton iteration. Theoretical proofs confirm that the Picard–Newton iteration is stable and achieves quadratic convergence, with a broader convergence basin than the usual Newton iteration, owing to its improved stability properties. Numerical experiments validate the theoretical conclusions and reveal that this method significantly outperforms standalone Picard or Newton iterations in several benchmark problems.