A posteriori error analysis of Banach spaces-based fully-mixed finite element methods for Boussinesq-type models

Abstract In this paper we consider Banach spaces-based fully-mixed variational formulations recently proposed for the Boussinesq and the Oberbeck–Boussinesq models, and develop reliable and efficient residual-based a posteriori error estimators for the 2D and 3D versions of the associated mixed finite element schemes. For the reliability analysis, we employ the global inf-sup condition for each sub-model, namely Navier–Stokes and heat equations in the case of Boussinesq, along with suitable Helmholtz decomposition in nonstandard Banach spaces, the approximation properties of the Raviart–Thomas and Clément interpolants, further regularity on the continuous solutions, and small data assumptions. In turn, the efficiency estimates follow from inverse inequalities and the localization technique through bubble functions in adequately defined local Lp spaces. Finally, several numerical results including natural convection in 3D differentially heated enclosures, are reported with the aim of confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithm.