In this paper we analyze the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem, where the viscosity of the fluid and the diffusion coefficient depend on the solution to the transport problem and its gradient, respectively. An augmented mixed variational formulation for both the fluid flow and the transport model is proposed. As a consequence, no discrete inf-sup conditions are required for the stability of the associated Galerkin scheme, and therefore arbitrary finite element subspaces can be used, which constitutes one of the main advantages of the present approach. In particular, the resulting fully-mixed finite element method can employ Raviart–Thomas spaces of order k for the Cauchy stress, continuous piecewise polynomials of degree \(k + 1\) for the velocity and for the scalar field, and discontinuous piecewise polynomial approximations for the gradient of the concentration. In turn, the Lax–Milgram lemma, monotone operators theory, and the classical Schauder and Brouwer fixed point theorems are utilized to establish existence of solution of the continuous and discrete formulations. In addition, suitable estimates, arising from the combined use of a regularity assumption with the Sobolev embedding and Rellich–Kondrachov compactness theorems, are also required for the continuous analysis. Then, sufficiently small data allow us to prove uniqueness of solution and to derive optimal a priori error estimates. Finally, several numerical tests, illustrating the performance of our method and confirming the predicted rates of convergence, are reported.

中文翻译：

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耦合流传输问题的增强全混合有限元方法

在本文中，我们分析了Stokes方程与由标量非线性对流扩散问题建模的输运问题的耦合，其中流体的粘度和扩散系数分别取决于输运问题及其梯度的解。提出了一种针对流体流动和运输模型的增强混合变分公式。结果，对于相关的Galerkin方案的稳定性，不需要离散的inf-sup条件，因此可以使用任意有限元子空间，这构成了本方法的主要优点之一。尤其是，所得的完全混合有限元方法可以对Cauchy应力采用阶数为k的Raviart-Thomas空间，即连续的分段多项式速度和标量场为\（k + 1 \），浓度梯度为不连续的分段多项式逼近。反过来，利用Lax–Milgram引理，单调算子理论以及经典的Schauder和Brouwer不动点定理来建立连续和离散公式解的存在性。此外，连续分析还需要将正则性假设与Sobolev嵌入和Rellich-Kondrachov紧致性定理结合使用而得出的合适估计值。然后，足够小的数据使我们能够证明解的唯一性并得出最佳的先验误差估计。最后，报告了一些数值试验，这些试验说明了我们方法的性能并确认了预测的收敛速度。