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-JosephSaffman law. The finite element subspaces consider Bernardi-Raugel and Raviart- KOFFI WILFRID HOUEDANOU et al. 38 Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. The a 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.

In this paper, we develop an a posteriori error analysis for the stationary Stokes-Darcy coupled problem approximated by conforming the finite element method on isotropic meshes in ℝ d , d ∈ 2 , 3 . The approach utilizes a new robust stabilized fully mixed discretization. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution plus the stabilization terms. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient.

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