In this work we present a new mixed finite element method for a class of steady-state natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid and a vector unknown involving the temperature, its gradient and the velocity. The introduction of these further unknowns lead to a mixed formulation where the aforementioned pseudostress tensor and vector unknown, together with the velocity and the temperature, are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the pseudostress tensor and the vector unknown, and discontinuous piece-wise polynomial elements of degree k for the velocity and temperature. With this choice of spaces, both, momentum and thermal energy, are conserved if the external forces belong to the velocity and temperature discrete spaces, respectively, which constitutes one of the main feature of our approach. We prove unique solvability for both, the continuous and discrete problems and provide the corresponding convergence analysis. Further variables of interest, such as the fluid pressure, the fluid vorticity, the fluid velocity gradient, and the heat-flux can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. Finally, several numerical results illustrating the performance of the method are provided.

在这项工作中，我们为一类稳态自然对流模型提供了一种新的混合有限元方法，该模型描述了受热源作用的非等温不可压缩流体的行为。我们的方法基于根据压力引入修改的伪应力张量以及流体和矢量的Navier–Stokes方程的扩散和对流项，其中矢量涉及温度，其梯度和速度。这些其他未知数的引入导致混合公式，其中上述伪应力张量和矢量未知数以及速度和温度是系统的主要未知数。然后可以通过使用度数为k的Raviart-Thomas元素定义相关的Galerkin方案对于伪应力张量和向量未知数，以及度数为k的不连续分段多项式元素速度和温度 通过这种空间选择，如果外力分别属于速度和温度离散空间，则动量和热能均得以保留，这是我们方法的主要特征之一。我们证明了连续和离散问题都具有独特的可解性，并提供了相应的收敛性分析。感兴趣的其他变量，例如流体压力，流体涡度，流体速度梯度和热通量，可以很容易地近似为具有相同收敛速度的有限元解的简单后处理。最后，提供了一些数值结果，说明了该方法的性能。