  In this paper, we first propose and analyze a novel mixed-type DG method for
the coupled Stokes-Darcy problem on simplicial meshes. The proposed formulation
is locally conservative. A mixed-type DG method in conjunction with the
stress-velocity formulation is employed for the Stokes equations, where the
symmetry of stress is strongly imposed. The staggered DG method is exploited to
discretize the Darcy equations. As such, the discrete formulation can be easily
adapted to account for the Beavers-Joseph-Saffman interface conditions without
introducing additional variables. Importantly, the continuity of normal
velocity is satisfied exactly at the discrete level. A rigorous convergence
analysis is performed for all the variables. Then we devise and analyze a
domain decomposition method via the use of Robin-type interface boundary
conditions, which allows us to solve the Stokes subproblem and the Darcy
subproblem sequentially with low computational costs. The convergence of the
proposed iterative method is analyzed rigorously. In particular, the proposed
iterative method also works for very small viscosity coefficient. Finally,
several numerical experiments are carried out to demonstrate the capabilities
and accuracy of the novel mixed-type scheme, and the convergence of the domain
decomposition method.
