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Pre-Publicación 2022-11

Sergio Caucao, Gabriel N. Gatica, Ricardo Oyarzúa, Paulo Zuñiga:

A posteriori error analysis of a mixed finite element method for the coupled Brinkman--Forchheimer and double-diffusion equations

Abstract:

In this paper we consider a partially augmented fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman--Forchheimer and double-diffusion equations, and develop an {\it a posteriori} error analysis for the $2$D and $3$D versions of the associated mixed finite element scheme. Indeed, we derive two reliable and efficient residual-based {\it a posteriori} error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity and inf-sup condition of the forms involved, a suitable assumption on the data, stable Helmholtz decompositions in Hilbert and Banach frameworks, and the local approximation properties of the Cl\'ement and Raviart--Thomas operators. 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 examples confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a 3D porous media with channel networks is considered.

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