  We consider in this paper, a new a posteriori residual type error estimator
of a conforming mixed finite element method for the coupling of fluid flow with
porous media flow on isotropic meshes. Flows are governed by the Navier-Stokes
and Darcy equations, respectively, and the corresponding transmission
conditions are given by mass conservation, balance of normal forces, and the
Beavers-Joseph-Saffman law. The finite element subspaces consider
Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise
constants for the pressures, and continuous piecewise linear elements for a
Lagrange multiplier defined on the interface. The posteriori error estimate is
based on a suitable evaluation on the residual of the finite element solution.
It is proven that the a posteriori error estimate provided in this paper is
both reliable and efficient. In addition, our analysis can be extended to other
finite element subspaces yielding a stable Galerkin scheme.
