In this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity and the temperature, we arrive at saddle point-type schemes in Banach spaces for both equations. In particular, and as suggested by the solvability of a related Neumann problem to be employed in the analysis, we need to make convenient choices of the Lebesgue and ${\textrm {H}}(div)$-type spaces to which the unknowns and test functions belong. The resulting coupled formulation is then written equivalently as a fixed-point operator, so that the classical Banach theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška theorem, suitable operators mapping Lebesgue spaces into themselves, regularity assumptions and the aforementioned Neumann problem, are employed to establish the unique solvability of the continuous formulation. Under standard hypotheses satisfied by generic finite element subspaces, the associated Galerkin scheme is analysed similarly and the Brouwer theorem yields existence of a solution. The respective a priori error analysis is also derived. Then, Raviart–Thomas elements of order $k\ge 0$ for the pseudoheat and the velocity and discontinuous piecewise polynomials of degree $\le k$ for the pressure and the temperature are shown to satisfy those hypotheses in the two-dimensional case. Several numerical examples illustrating the performance and convergence of the method are reported, including an application into the equivalent problem of miscible displacement in porous media.

中文翻译：

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基于 Lp 空间的公式为耦合的达西和热方程产生了一种新的完全混合有限元方法

在这项工作中，我们提出并分析了一种新的完全混合有限元方法，用于解决由达西和热方程耦合给出的非线性问题。除了流体的速度、压力和温度变量之外，我们的方法是基于引入伪热通量作为进一步的未知数。作为它的结果，并且由于涉及速度和温度的对流项，我们在巴拿赫空间中得到了两个方程的鞍点型方案。特别是，正如在分析中使用的相关 Neumann 问题的可解性所建议的那样，我们需要方便地选择 Lebesgue 和 ${\textrm {H}}(div)$-type 空间，其中未知数和测试功能属于。然后将得到的耦合公式等效地写为定点算子，因此经典的巴纳赫定理，结合相应的巴布什卡-布雷齐理论、巴纳赫-内卡斯-巴布什卡定理、将勒贝格空间映射到自身的合适算子、正则性假设和上述诺伊曼问题，被用来建立连续的唯一可解性公式。在通用有限元子空间满足的标准假设下，类似地分析了相关的Galerkin格式，并且Brouwer定理产生了解的存在性。还导出了相应的先验误差分析。然后，Raviart-Thomas 阶 $k\ge 0$ 的伪热和速度和不连续分段多项式 $\le k$ 度的压力和温度被证明在二维情况下满足这些假设。