A Posteriori Error Analysis of a Mixed-Primal Finite Element Method for the Boussinesq Problem with Temperature-Dependent Viscosity

Abstract

We have recently proposed a new finite element method for a more general Boussinesq model in 2D given by the case in which the viscosity of the fluid depends on its temperature. Our approach is based on a pseudostress–velocity–vorticity mixed formulation for the momentum equations, which is suitably augmented with Galerkin-type terms, coupled with the usual primal formulation for the energy equation, along with the introduction of the normal heat flux on the boundary as a Lagrange multiplier taking care of the fact that the prescribed temperature there becomes an essential condition. Then, fixed-point arguments using Banach and Brouwer theorems, in addition to other classical tools from functional and numerical analysis, provide sufficient conditions ensuring well-posedness of the resulting continuous and discrete sytems, together with the corresponding error estimates and associated rates of convergence. In the present work we complement these results with the derivation of a reliable and efficient residual-based a posteriori error estimator for the aforementioned augmented mixed-primal finite element method. Duality techniques, Helmholtz decompositions, and the approximation properties of the Raviart–Thomas and Clement interpolants are applied to obtain a reliable global error indicator. In turn, standard tools including the usual localization technique of bubble functions and inverse inequalities, and a regularity assumption originally utilized in the previous well-posedness and a priori error analyses, are employed to prove its efficiency. In both cases, reliability and efficiency, the estimates are shown at global level. Finally, a reliable fully local and computable a posteriori error estimator induced by the aforementioned one is deduced, and several numerical results illustrating its performance and validating the expected behaviour of the associated adaptive algorithm are reported.