An Augmented Lagrangian Preconditioner for the Magnetohydrodynamics Equations at High Reynolds and Coupling Numbers

The magnetohydrodynamics (MHD) equations are generally known to be difficult solve numerically, due their highly nonlinear structure and the strong coupling between electromagnetic hydrodynamic variables, especially for high Reynolds numbers. In this work, we present a scalable augmented Lagrangian preconditioner finite element discretization of $B$-$E$ formulation incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect numbers in two dimensions good results three dimensions. We extend method fully implicit methods time-dependent problems which robustly both Our approach relies on specialized parameter-robust multigrid blocks. scheme ensures exactly divergence-free approximations velocity magnetic field up tolerances. confirm robustness by numerical experiments consider fluid 10,000 100,000 transient