Abstract In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n-dimensional fluids, n ∈ { 2 , 3 } {n\in\{2,3\}} , with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.

中文翻译：

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具有温度相关参数的 n 维 Boussinesq 问题的全混合有限元方法

摘要 在本文中，我们介绍并分析了一种用于 n 维流体自由对流的高阶全混合有限元方法，n ∈ { 2 , 3 } {n\in\{2,3\}} ，具有随温度变化的粘度和热导率。数学模型由连续性方程、动量方程 (Navier-Stokes) 和能量通过 Boussinesq 近似以及混合热边界条件和速度的 Dirichlet 条件耦合给出。由于与流体特性的温度有关，因此定义了几个额外的变量，从而产生了寻求应变率、伪应力和涡度张量、速度、温度梯度和伪热矢量以及流体温度的增强公式。使用定点方法，精确解的小数据假设和稍高的正则性假设在连续和离散水平上都提供了必要的适定结果。此外，作为增强的结果，Galerkin 方案的适定性不需要离散的 inf-sup 条件，这提供了关于有限元空间的选择自由。特别是，我们建议基于 Raviart-Thomas、拉格朗日和不连续元素的组合，我们为其推导出最佳先验误差估计。最后，报告了几个说明该方法性能和确认理论收敛速度的数值例子。并且作为增强的结果，Galerkin 方案的适定性不需要离散的 inf-sup 条件，这提供了关于有限元空间的选择自由。特别是，我们建议基于 Raviart-Thomas、拉格朗日和不连续元素的组合，我们为其推导出最佳先验误差估计。最后，报告了几个说明该方法性能和确认理论收敛速度的数值例子。并且作为增强的结果，Galerkin 方案的适定性不需要离散的 inf-sup 条件，这提供了关于有限元空间的选择自由。特别是，我们建议基于 Raviart-Thomas、拉格朗日和不连续元素的组合，我们为其推导出最佳先验误差估计。最后，报告了几个说明该方法性能和确认理论收敛速度的数值例子。