In this paper we propose and analyse a new formulation and pointwise divergence-free mixed finite element methods for the numerical approximation of Darcy--Brinkman equations in vorticity--velocity--pressure form, coupled with a transport equation for thermal energy with viscous dissipative effect and mixed Navier-type boundary conditions. The solvability analysis of the continuous and discrete problems is significantly more involved than usual as it hinges on Banach spaces needed to properly control the advective and dissipative terms in the non-isothermal energy balance equation. We proceed by decoupling the set of equations and use the Banach fixed-point theorem in combination with the abstract theory for perturbed saddle-point problems. Some of the necessary estimates are straightforward modifications of well-known results, while other technical tools require a more elaborated analysis. The velocity is approximated by Raviart--Thomas elements, the vorticity uses N\'ed\'elec spaces of the first kind, the pressure is approximated by piecewise polynomials, and the temperature by continuous and piecewise polynomials of one degree higher than pressure. Special care is needed to establish discrete inf-sup conditions since the curl of the discrete vorticity is not necessarily contained in the discrete velocity space, therefore suggesting to use two different Raviart--Thomas interpolants. A discrete fixed-point argument is used to show well-posedness of the Galerkin scheme. Error estimates in appropriate norms are derived, and a few representative numerical examples in 2D and 3D and with mixed boundary conditions are provided.

翻译：本文提出并分析了一种新的表述方法及点态无散度混合有限元方法，用于以涡度-速度-压力形式表达的Darcy-Brinkman方程的数值逼近，该方法耦合了考虑粘性耗散效应的热输运方程及混合Navier型边界条件。由于连续与离散问题的可解性分析依赖于Banach空间（该空间需恰当控制非等温能量平衡方程中的对流项与耗散项），其复杂程度显著高于常规情况。我们通过解耦方程组，结合Banach不动点定理与扰动鞍点问题的抽象理论进行分析。部分必要估计是经典结果的直接推广，而其他技术工具则需要更精细的分析。速度采用Raviart-Thomas元逼近，涡度采用第一类Nédélec空间，压力采用分片多项式逼近，温度采用比压力高一阶的连续分片多项式逼近。由于离散涡度的旋度不一定包含于离散速度空间，需特别注意建立离散inf-sup条件，因此建议采用两种不同的Raviart-Thomas插值算子。通过离散不动点论证证明了Galerkin格式的适定性。推导了相应范数下的误差估计，并提供了二维与三维情形下具有混合边界条件的若干代表性数值算例。