Unconditionally stable operator splitting algorithms for the incompressible magnetohydrodynamics (MHD) system discretized by a stabilized finite element formulation based on projections

Abstract

In this article, we propose different splitting procedures for the transient incompressible magnetohydrodynamics (MHD) system that are unconditionally stable. We consider two levels of splitting, on one side we perform the segregation of the fluid pressure and magnetic pseudo‐pressure from the vectorial fields computation. At the second level, the fluid velocity and induction fields are also decoupled. This way, we transform a fully coupled indefinite multi‐physics system into a set of smaller definite ones, clearly reducing the CPU cost. With regard to the finite element approximation, we stick to an unconditionally convergent stabilized finite element formulation because it introduces convection stabilization, allows to circumvent inf‐sup conditions (clearly simplifying implementation issues), and is able to capture non‐smooth solutions of the magnetic subproblem. However, residual‐based finite element formulations are not suitable for segregation, because they lose the skew‐symmetry of the off‐diagonal blocks. Therefore, in this work, we have proposed a novel term‐by‐term stabilization of the MHD system based on projections that is still unconditionally convergent.