A posteriori error analysis of a mixed finite element method for the stationary convective Brinkman–Forchheimer problem

Abstract

We consider a Banach spaces-based mixed variational formulation that has been recently proposed for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations, and develop a reliable and efficient residual-based a posteriori error estimator for the 2D and 3D versions of the associated mixed finite element scheme. For the reliability analysis, we utilize the global inf-sup condition of the problem, combined with appropriate small data assumptions, a stable Helmholtz decomposition in nonstandard Banach spaces, and the local approximation properties of the Raviart–Thomas and Clément interpolants. In turn, inverse inequalities, the localization technique based on bubble functions in local $L^p$-spaces, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported. In particular, the case of flow through a 2D porous medium with fracture networks is considered.