Volume 55, Number 2, March-April 2021
|Page(s)||659 - 687|
|Published online||31 March 2021|
Residual-based a posteriori error analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations
Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile
2 CI 2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
3 GIMNAP-Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción and CI 2MA, Universidad de Concepción, Casilla 160-C Concepción Chile
* Corresponding author: firstname.lastname@example.org
Accepted: 25 January 2021
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 Lp 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.
Mathematics Subject Classification: 65N30 / 65N12 / 65N15 / 35Q79 / 80A20 / 76R05 / 76D07
Key words: Navier–Stokes problem / Darcy–Forchheimer problem / primal-mixed finite element methods / a posteriori error analysis
© EDP Sciences, SMAI 2021
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