  We present a detailed comparison of several recent and new approaches to
multigrid solver algorithms suitable for the solution of 5d chiral fermion
actions such as Domain Wall fermions in the Shamir formulation, and also for
the Partial Fraction and Continued Fraction overlap. Our focus is on the
acceleration of gauge configuration sampling, and a compact nearest neighbour
stencil is required to limit the calculational cost of obtaining a coarse
operator. This necessitates the coarsening of a nearest neighbour operator to
preserve sparsity in coarsened grids, unlike HDCG. We compare the approaches of
HDCR and the Multigrid algorithm and also several new hybrid schemes. In this
work we introduce a new recursive Chebyshev polynomial based setup scheme. We
find that the HDCR approach, can both setup, and solve standard Shamir Domain
Wall Fermions faster than a single solve with red-black preconditioned
Conjugate Gradients on large volumes and for modern GPU systems such as the
Summit supercomputer. This is promising for the acceleration of HMC,
particularly if setup costs are shared across multiple Hasenbusch determinant
factors. The setup scheme is likely generally applicable to other Fermion
actions.
