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## Sergio Caucao, Gabriel N. Gatica, Juan P. Ortega:

### Abstract:

In this paper we consider a Banach spaces-based fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman--Forchheimer and double-diffusion equations, and develop a reliable and efficient residual-based {\it a posteriori} error estimator for the $2$D and $3$D versions of the associated mixed finite element scheme. For the reliability analysis, we employ the strong monotonicity and inf-sup conditions of the operators involved, along with a suitable assumption on the data, stable Helmholtz decomposition in nonstandard Banach spaces, and local approximation properties of the Raviart--Thomas and Cl\'ement interpolants. In turn, inverse inequalities, the localization technique through bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a $2$D porous media with channel networks is considered.

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