  We study the impact of spatial coarsening on the convergence of the Parareal
algorithm, both theoretically and numerically. For initial value problems with
a normal system matrix, we prove a lower bound for the Euclidean norm of the
iteration matrix. When there is no physical or numerical diffusion, an
immediate consequence is that the norm of the iteration matrix cannot be
smaller than unoty as soon as the coarse problem has fewer degrees-of-freedom
than the fine. This prevents a theoretical guarantee for monotonic convergence,
which is necessary to obtain meaningful speedups. For diffusive problems, in
the worst-case where the iteration error contracts only as fast as the powers
of the iteration matrix norm, making Parareal as accurate as the fine method
will take about as many iterations as there are processors, making meaningful
speedup impossible. Numerical examples with a non-normal system matrix show
that for diffusive problems good speedup is possible, but that for
non-diffusive problems the negative impact of spatial coarsening on convergence
is big.
