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Pre-Publicación 2016-37

Eligio Colmenares, Gabriel N. Gatica, Ricardo Oyarzua:

A posteriori error analysis of an augmented mixed-primal formulation for the stationary Boussinesq model

Abstract:

In an earlier work of us, a new mixed finite element scheme was developed for the Boussinesq model describing natural convection. Our methodology consisted of a fixed-point strategy for the variational problem that resulted after introducing a modified pseudostress tensor and the normal component of the temperature gradient as auxiliary unknowns in the corresponding Navier--Stokes and advection--diffusion equations defining the model, respectively, along with the incorporation of parameterized redundant Galerkin terms. The well--posedness of both the continuous and discrete settings, the convergence of the associated Galerkin scheme, as well as a priori error estimates of optimal order were stated there. In this work we complement the numerical analysis of our aforementioned augmented mixed--primal method by carrying out a corresponding a posteriori error estimation in two and three dimensions. Standard arguments relying on duality techniques, and suitable Helmholtz decompositions are used to derive a global error indicator and to show its reliability. A globally efficiency property with respect to the natural norm is further proved via usual localization techniques of bubble functions. Finally, an adaptive algorithm based on a reliable, fully local and computable a posteriori error estimator induced by the aforementioned one is proposed, and its performance and effectiveness are illustrated through a few numerical examples in two dimensions.

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Esta prepublicacion dio origen a la(s) siguiente(s) publicación(es) definitiva(s):

Eligio COLMENARES, Gabriel N. GATICA, Ricardo OYARZUA: A posteriori error analysis of an augmented mixed-primal formulation for the stationary Boussinesq model. Calcolo, vol. 54, 3, pp. 1055-1095, (2017).