In this paper we propose a new mixed-primal formulation for heat-driven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. We analyze the well-posedness of the governing equations in this mathematical structure, for which we employ the Banach fixed-point theorem and the generalized theory of saddle-point problems. The motivation is to overcome a drawback in a recent work by the authors where, in the mixed formulation for the momentum equation, the reciprocal of the viscosity is a pre-factor to a tensor product of velocities; making the analysis quite restrictive, as one needs a given continuous injection that holds only in 2D. We show in this work that by adding both the pseudo-stress and the strain rate tensors as new unknowns in the problem, we get more flexibility in the analysis, covering also the 3D case. The rest of the formulation is based on eliminating the pressure, incorporating augmented Galerkin-type terms in the mixed form of the momentum equation, and defining the normal heat flux as a suitable Lagrange multiplier in a primal formulation for the energy equation. Additionally, the symmetry of the stress is imposed in an ultra-weak sense, and consequently the vorticity tensor is no longer required as part of the unknowns. A finite element method that follows the same setting is then proposed, where we remark that both pressure and vorticity can be recovered from the principal unknowns via postprocessing formulae. The solvability of the discrete problem is analyzed by means of the Brouwer fixed-point theorem, and we derive error estimates in suitable norms. Numerical examples illustrate the performance of the new schem and its use in the simulation of mantle convection, and they also confirm the theoretical rates of convergence.

中文翻译：

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随温度变化的n维Boussinesq问题的一种新的混合有限元方法

在本文中，我们提出了一种新的混合主方程式，该方程由稳态的Boussinesq方程建模，具有随温度变化的粘度，用于热流。我们在这种数学结构中分析了控制方程的适定性，为此我们采用了Banach不动点定理和鞍点问题的广义理论。动机是克服作者最近的工作中的一个缺点，即在动量方程的混合公式中，粘度的倒数是速度张量积的前因。由于需要给定的仅在2D模式下进行的连续注入，因此分析变得十分严格。我们在这项工作中表明，通过将伪应力和应变率张量都添加为问题中的新未知数，我们可以在分析中获得更大的灵活性，还涵盖了3D外壳。该公式的其余部分基于消除压力，在动量方程的混合形式中合并增强的Galerkin型项以及将常规热通量定义为能量公式的原始公式中的合适Lagrange乘数。另外，应力的对称性是在超弱的意义上施加的，因此不再需要涡度张量作为未知量的一部分。然后提出了遵循相同设置的有限元方法，在此我们指出可以通过后处理公式从主要未知数中恢复压力和涡度。离散问题的可解性通过Brouwer不动点定理进行分析，我们得出了适合范数的误差估计。