  NOTE: this is a shortened version of the abstract of the paper.
  Multigrid methods for propagators in gauge fields are investigated.
  Gauge fields are incorporated in algorithms in a covariant way. This avoids
the necessity for gauge fixing in computations of propagators.
  The kernel $C$ of the restriction operator which averages from one grid to
the next coarser grid is defined by projection on the ground- state of a local
Hamiltonian. The idea behind this definition is that the appropriate notion of
smoothness depends on the dynamics. The ground-state projection choice of $C$
is usable in arbitrary space-time dimension $d$ and for arbitrary gauge group.
We discuss proper averaging operations for bosons and for staggered fermions.
The averaging kernels $C$ can be used not only in deterministic multigrid
computations, but also in multigrid Monte Carlo simulations, and for the
definition of block spins and blocked gauge fields in Monte Carlo
renormalization group studies of gauge theories. Actual numerical computations
of kernels and propagators are performed in compact four-dimensional $SU(2)$
gauge fields. A central result of the present work is that {\em the multigrid
method works in arbitrarily disordered gauge fields, in principle\/}.
