  We consider the time-dependent Stokes-Darcy problem as a model case for the
challenges involved in solving coupled systems. Keeping the model, its
discretization, and the underlying numerics for the subproblems in the
free-flow domain and the porous medium constant, we focus on different solver
approaches for the coupled problem. We compare a partitioned coupling approach
using the coupling library preCICE with a monolithic block-preconditioned one
that is tailored to different formulations of the problem. Both approaches
enable the reuse of already available iterative solvers and preconditioners, in
our case, from the DuMux framework. Our results indicate that the approaches
can yield performance and scalability improvements compared to using direct
solvers: Partitioned coupling is able to solve large problems faster if
iterative solvers with suitable preconditioners are applied for the
subproblems. The monolithic approach shows even stronger requirements on
preconditioning, as standard simple solvers fail to converge. Our monolithic
block preconditioning yields the fastest runtimes for large systems, but they
vary strongly with the preconditioner configuration. Interestingly, using a
specialized Uzawa preconditioner for the Stokes subsystem leads to overall
increased runtimes compared to block preconditioners utilizing a more general
algebraic multigrid. This highlights that optimizing for the non-coupled cases
does not always pay off.
