  The parallel full approximation scheme in space and time (PFASST) introduced
by Emmett and Minion in 2012 is an iterative strategy for the temporal
parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is
similar in spirit to a space-time FAS multigrid method performed over multiple
time-steps in parallel. However, since the original focus of PFASST has been on
the performance of the method in terms of time parallelism, the solution of any
spatial system arising from the use of implicit or semi-implicit temporal
methods within PFASST have simply been assumed to be solved to some desired
accuracy completely at each sub-step and each iteration by some unspecified
procedure. It hence is natural to investigate how iterative solvers in the
spatial dimensions can be interwoven with the PFASST iterations and whether
this strategy leads to a more efficient overall approach. This paper presents
an initial investigation on the relative performance of different strategies
for coupling PFASST iterations with multigrid methods for the implicit
treatment of diffusion terms in PDEs. In particular, we compare full accuracy
multigrid solves at each sub-step with a small fixed number of multigrid
V-cycles. This reduces the cost of each PFASST iteration at the possible
expense of a corresponding increase in the number of PFASST iterations needed
for convergence. Parallel efficiency of the resulting methods is explored
through numerical examples.
