Analysis of the Picard-Newton finite element iteration for the stationary incompressible inductionless MHD equations

In this paper, we propose and analyze the Picard-Newton finite element iteration for the stationary incompressible inductionless magnetohydrodynamics (MHD) equations. In finite element discretization, the hydrodynamic unknowns are approximated by stable finite element pairs, and the electromagnetic system is discretized by using the face-volume pairs. To solve the nonlinear discretized problem efficiently, our method consists of first applying the Picard iteration and then applying the Newton iteration. The Picard-Newton iteration is proved to be globally stable under the uniqueness condition and quadratically convergent under the stronger uniqueness condition. Thanks to the improved stability property, this solver has a larger convergence basin than the usual Newton iteration. Numerical tests confirm our theoretical analysis and show that the Picard-Newton iteration dramatically excels both the Picard and Newton iterations in several benchmark problems.