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

¹ 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: scaucao@ucsc.cl

Received: 20 April 2020
Accepted: 25 January 2021

Abstract

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^p 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.