A posteriori error analysis of a fully-mixed formulation for the Navier-Stokes/Darcy coupled problem with nonlinear viscosity

In this paper we consider an augmented fully-mixed variational formulation that has been recently proposed for the coupling of the Navier-Stokes equations (with nonlinear viscosity) and the linear Darcy model, and derive a reliable and efficient residual-based a posteriori error estimator for the associated mixed finite element scheme. The finite element subspaces employed are piecewise constants, Raviart-Thomas elements of lowest order, continuous piecewise linear elements, and piecewise constants for the strain, Cauchy stress, velocity, and vorticity in the fluid, respectively, whereas Raviart-Thomas elements of lowest order for the velocity, piecewise constants for the pressure, and continuous piecewise linear elements for the traces, are considered in the porous medium. The proof of reliability of the estimator relies on a global inf-sup condition, suitable Helmholtz decompositions in the fluid and the porous medium, the local approximation properties of the Clément and Raviart-Thomas operators, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.