A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

Abstract

In this article, we consider an augmented fully mixed variational formulation that has been recently proposed for the nonisothermal Oldroyd–Stokes problem, and develop an a posteriori error analysis for the 2‐D and 3‐D versions of the associated mixed finite element scheme. More precisely, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem on arbitrary (convex or nonconvex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the bilinear forms of the continuous formulation, suitable assumptions on the domain and the data, stable Helmholtz decompositions, and the local approximation properties of the Clément and Raviart–Thomas operators. On the other hand, inverse inequalities, the localization technique based on bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the a posteriori error estimators and illustrating the performance of the associated adaptive algorithms are reported.

A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media

“…In this section we follow to [3, 4, 13, 14], and [15], and introduce an alternative a posteriori error estimator for the scheme (3.21a) and (3.21b) which is obtained after performing a different algebraic manipulation of the term $$$ \left\Vert {\mathcal{R}}^{\mathtt{Flow}}\right\Vert $$$ in (4.29) (cf. Lemma 4.4) allowing us to prove the respective reliability property without owing to a Helmholtz decomposition, in contrast to the previous one $$$ \boldsymbol{\Theta} $$$.…”