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Pre-Publicación 2019-33

Sergio Caucao, Gabriel N. Gatica, Ricardo Oyarzúa, Felipe Sandoval:

Residual-based a posteriori error analysis for the coupling of the Navier-Stokes and Darcy-Forchheimer equations


In this paper we consider two mixed variational formulations that have been recently proposed for the coupling of the Navier-Stokes and Darcy-Forchheimer equations, and derive reliable and efficient residual-based a posteriori error estimators suitable for adaptive mesh-refinement methods. For the reliability analysis of both schemes we make use of the inf-sup condition and the strict monotonicity of the operators involved, suitable Helmholtz decomposition in nonstandard Banach space in the porous medium, local approximation properties of the Clement interpolant and Raviart-Thomas operator, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on triangle-buble and edge-buble functions in local L^p spaces, are the main tools for study the efficiency estimate. In addition, for one of the schemes, we derive two estimators, one obtained as a direct consequence of the Cauchy-Schwarz inequality and the other one employing a Helmholtz decomposition. Finally, several numerical results confirming the properties of the estimators and illustrating the performance of the associated adaptive algorithm are reported.

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