In this work we present and analyze a fully‐mixed formulation for the nonlinear model given by the coupling of the Navier–Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. Our approach yields non‐Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier–Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well‐known Schauder and Banach theorems, combined with classical results on nonlinear monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ≥ 0, Raviart–Thomas spaces of order k, continuous piecewise polynomials of degree ≤k + 1 and piecewise polynomials of degree ≤k are employed in the fluid for approximating the pseudostress tensor, velocity and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ≤k for the velocity and pressure, constitute a feasible choice in the porous medium. A priori error estimates and associated rates of convergence are derived, and several numerical examples illustrating the good performance of the method are reported.

中文翻译：

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Navier–Stokes和Darcy–Forchheimer方程耦合的完全混合有限元方法

在这项工作中，我们提出并分析了由Navier–Stokes和Darcy–Forchheimer方程与界面上的Beavers–Joseph–Saffman条件耦合而给出的非线性模型的完全混合公式。我们的方法为相应的算子方程产生非希尔伯特范数空间和双重鞍点结构。此外，由于Navier–Stokes方程中的对流项迫使速度生活在比平常更小的空间中，因此我们使用合适的Galerkin类型项扩展了变分公式。然后将所得的扩充方案等效地写为一个不动点方程，以便将著名的Schauder和Banach定理与非线性单调算子的经典结果结合起来，以证明连续和离散系统的独特可解性。尤其是，ķ  ≥0 ，订单的Raviart -托马斯空间ķ，程度的连续分段多项式≤ ķ  + 1次的和分段多项式≤ ķ在流体被用于分别近似pseudostress张量，速度和涡度，而Raviart -托马斯空间顺序的ķ和度的分段多项式≤ ķ的速度和压力，在构成所述多孔介质中的可行的选择。推导了先验误差估计和相关的收敛速度，并报告了一些数值实例，说明了该方法的良好性能。