  We present a scalable block preconditioning strategy for the trace system
coming from the high-order hybridized discontinuous Galerkin (HDG)
discretization of incompressible resistive magnetohydrodynamics (MHD). We
construct the block preconditioner with a least squares commutator (BFBT)
approximation for the inverse of the Schur complement that segregates out the
pressure unknowns of the trace system. The remaining velocity, magnetic field,
and Lagrange multiplier unknowns form a coupled nodal unknown block (the upper
block), for which a system algebraic multigrid (AMG) is used for the
approximate inverse. The complexity of the MHD equations together with the
algebraic nature of the statically condensed HDG trace system makes the choice
of smoother in the system AMG part critical for the convergence and performance
of the block preconditioner. Our numerical experiments show GMRES
preconditioned by ILU(0) of overlap zero as a smoother inside system AMG
performs best in terms of robustness, time per nonlinear iteration and memory
requirements. With several transient test cases in 2D and 3D including the
island coalescence problem at high Lundquist number we demonstrate the
robustness and parallel scalability of the block preconditioner. Additionally
for the upper block a preliminary study of an alternate nodal block system
solver based on a multilevel approximate nested dissection is presented. On a
2D island coalescence problem the multilevel approximate nested dissection
preconditioner shows better scalability with respect to mesh refinement than
the system AMG, but is relatively less robust with respect to Lundquist number
scaling.
