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A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes-Darcy coupled problem. (English) Zbl 1228.76085
Summary: In this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the 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. We consider dual-mixed formulations in both media, which yields the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the traces of the porous media pressure and the fluid velocity on the interface, as the resulting unknowns. The set of feasible finite element subspaces includes Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the traces. We derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. The proof of reliability makes use of the global inf-sup condition, Helmholtz decompositions in both media, and local approximation properties of the Clément interpolant and Raviart-Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 65N15 Error bounds for boundary value problems involving PDEs
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