  For time-dependent partial differential equations, parallel-in-time
integration using the "parallel full approximation scheme in space and time"
(PFASST) is a promising way to accelerate existing space-parallel approaches
beyond their scaling limits. Inspired by the classical Parareal method and
multigrid ideas, PFASST allows to integrate multiple time-steps simultaneously
using a space-time hierarchy of spectral deferred correction sweeps. While many
use cases and benchmarks exist, a solid and reliable mathematical foundation is
still missing. Very recently, however, PFASST for linear problems has been
identified as multigrid method and in this paper, we will use this multigrid
formulation and in particular PFASST's iteration matrix to show that in the
non-stiff as well as in the stiff limit PFASST indeed is a convergent iterative
method. We will provide upper bounds for the spectral radius of the iteration
matrix and investigate how PFASST performs for increasing numbers of parallel
time-steps. Finally, we will demonstrate that the results obtained here indeed
relate to actual PFASST runs.
