  The aim of this paper is to propose a systematic way to obtain convergent
finite element schemes for the Darcy-Stokes flow problem by combining
well-known mixed finite elements that are separately convergent for Darcy and
Stokes problems. In the approach in which the Darcy problem is set in its
natural $\mathbf{H}(\text{div})$ formulation and the Stokes problem is
expressed in velocity-pressure form, the transmission condition ensuring global
mass conservation becomes essential. As opposed to the strategy that handles
weakly this transmission condition through a Lagrange multiplier, we impose
here this restriction exactly in the space of global velocity field. Our
analysis of the Galerkin discretization of the resulting problem reveals that,
if the mixed finite element space used in the Darcy domain admits an
$\mathbf{H}(\text{div})$-stable discrete lifting of the normal trace, then it
can be combined with any stable Stokes mixed finite element of the same order
to deliver a stable global method with quasi-optimal convergence rate. Finally,
we present a series of numerical tests confirming our theoretical convergence
estimates.
