We consider the coupled Navier–Stokes/transport equations with nonlinear transmission conditions, which constitute one of the most common models utilized to simulate a reverse osmosis effect in water desalination processes when considering feed and permeate channels coupled through a semi-permeate membrane. The variational formulation consists of a set of equations where the velocities, the concentrations, along with tensors and vector fields introduced as auxiliary unknowns and two Lagrange multipliers are the main unknowns of the system. The latter are introduced to deal with the trace of functions that do not have enough regularity to be restricted to the boundary. In addition, the pressures can be recovered afterwards by a postprocessing formula. As a consequence, we obtain a nonlinear Banach spaces-based mixed formulation, which has a perturbed saddle point structure. We analyze the continuous and discrete solvability of this problem by linearizing the perturbation and applying the classical Banach fixed point theorem along with the Banach–Nečas–Babuška result. Regarding the discrete scheme, feasible choices of finite element subspaces that can be used include Raviart–Thomas spaces for the auxiliary tensor and vector unknowns, piecewise polynomials for the velocities and the concentrations, and continuous polynomial space of lowest order for the traces, yielding stable discrete schemes. An optimal a priori error estimate is derived, and numerical results illustrating both, the performance of the scheme confirming the theoretical rates of convergence, and its applicability, are reported.

中文翻译：

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一种用于模拟反渗透过程的 Navier-Stokes/运输系统耦合的符合混合有限元方法


我们考虑了具有非线性传输条件的 Navier-Stokes/传输耦合方程，当考虑通过半渗透膜耦合的进料和渗透通道时，它构成了用于模拟海水淡化过程中反渗透效应的最常见模型之一。变分公式由一组方程组成，其中速度、浓度以及作为辅助未知数引入的张量和向量场和两个拉格朗日乘子是系统的主要未知数。引入后者是为了处理没有足够规律性而被限制在边界的函数的跟踪。此外，压力可以在之后通过后处理公式恢复。因此，我们得到了一个非线性的 Banach 空间混合公式，它具有扰动鞍点结构。我们通过线性化扰动并应用经典的 Banach 不动点定理以及 Banach-Nečas-Babuška 结果来分析这个问题的连续和离散可解性。关于离散方案，可以使用的有限元子空间的可行选择包括用于辅助张量和向量未知数的 Raviart-Thomas 空间、用于速度和浓度的分段多项式以及用于轨迹的最低阶的连续多项式空间，从而产生稳定的离散方案。推导出了最佳的先验误差估计，并报告了说明两者的数值结果，确认了理论收敛率的方案的性能及其适用性。