In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier–Stokes and Darcy–Forchheimer equations, and derive, though in a non-standard sense, a reliable and efficient residual-based a posteriori error estimator suitable for an adaptive mesh-refinement method. For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Clément interpolant and Raviart–Thomas operator, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in local L$$ spaces, are the main tools for developing the efficiency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.
Keywords: Navier–Stokes problem, Darcy–Forchheimer problem, primal-mixed finite element methods, $$ error analysis
@article{M2AN_2021__55_2_659_0,
author = {Caucao, Sergio and Gatica, Gabriel N. and Oyarz\'ua, Ricardo and Sandoval, Felipe},
title = {Residual-based \protect\emph{a posteriori} error analysis for the coupling of the {Navier{\textendash}Stokes} and {Darcy{\textendash}Forchheimer} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {659--687},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {2},
doi = {10.1051/m2an/2021005},
mrnumber = {4238779},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021005/}
}
TY - JOUR AU - Caucao, Sergio AU - Gatica, Gabriel N. AU - Oyarzúa, Ricardo AU - Sandoval, Felipe TI - Residual-based a posteriori error analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 659 EP - 687 VL - 55 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021005/ DO - 10.1051/m2an/2021005 LA - en ID - M2AN_2021__55_2_659_0 ER -
%0 Journal Article %A Caucao, Sergio %A Gatica, Gabriel N. %A Oyarzúa, Ricardo %A Sandoval, Felipe %T Residual-based a posteriori error analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 659-687 %V 55 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021005/ %R 10.1051/m2an/2021005 %G en %F M2AN_2021__55_2_659_0
Caucao, Sergio; Gatica, Gabriel N.; Oyarzúa, Ricardo; Sandoval, Felipe. Residual-based a posteriori error analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 659-687. doi: 10.1051/m2an/2021005
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