In this work, a parallel-in-time method is combined with a multigrid algorithm
and further on with a spatial coarsening strategy. The most famous parallel-in-time
method is the parareal algorithm. Depending on two different operators, it enables the
parallelism of time-dependent problems. The operator with huge effort is carried out
in parallel. But despite parallelization this can lead to long run times for long-term
problems. Since the parareal algorithm has a two-level structure and the time-parallel
multigrid methods are also widespread in the area of parallel time integration, we
combine these approaches. We use the parareal algorithm as a smoothing operator in
the basic framework of a geometrical multigrid method, where we apply a coarsening
strategy in time. So we get a multigrid in time method which is strongly parallelizable.
For partial differential equations we add an extra spatial coarsening strategy to our
multigrid parareal version. All in all we get a method, which has a high parallel
efficiency and converges fast due to the multigrid framework, which is shown in the
numerical studies of this work. So we will get a highly accurate solution and can greatly
reduce the parallel complexity, which is especially important for long-term problems
with a limited number of processors
