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.

中文翻译：

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Boussinesq 型模型的基于 Banach 空间的全混合有限元方法的后验误差分析*

在本文中，我们考虑了最近为 Boussinesq 和 Oberbeck-Boussinesq 模型提出的基于 Banach 空间的完全混合变分公式，并为相关混合有限元的 2D 和 3D 版本开发了可靠且有效的基于残差的后验误差估计器计划。对于可靠性分析，我们对每个子模型采用全局 inf-sup 条件，即在 Boussinesq 的情况下的 Navier-Stokes 和热方程，以及非标准 Banach 空间中合适的 Helmholtz 分解，Raviart-Thomas 的近似属性和 Clément 插值，连续解的进一步规律性和小数据假设。反过来，效率估计来自反不等式和通过气泡函数在充分定义的局部 L 中的定位技术p空格。最后，报告了几个数值结果，包括 3D 不同加热外壳中的自然对流，目的是确认估计器的理论特性并说明相关自适应算法的性能。