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.
This preprint gave rise to the following definitive publication(s):
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. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 55, 2, pp. 659-687, (2021).