We propose and analyze an augmented mixed finite element method for the pseudostress-velocity formulation of the stationary convective Brinkman-Forchheimer problem in $\mathrm{R}^d$, $d\in \{2,3\}$. Since the convective and Forchheimer terms 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 written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart-Thomas spaces of order $k\geq 0$ for the pseudostress tensor and continuous piecewise polynomials of degree $\le k + 1$ for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Cl\'ement and Raviart-Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behaviour of the associated adaptive algorithms, are reported. In particular, the case of flow through a $2$D porous media with fracture networks is considered.

翻译：我们针对$\mathrm{R}^d$（$d\in \{2,3\}$）中稳态对流Brinkman-Forchheimer问题的拟应力-速度形式，提出并分析一种增强型混合有限元方法。由于对流项和Forchheimer项迫使速度场处于比常规更小的函数空间，我们通过引入合适的Galerkin型项来增强变分形式。所得增强格式等价地表达为不动点方程，从而结合Schauder与Banach不动点定理及Lax-Milgram定理，可证明连续问题的解存在唯一性。有限元离散采用阶数为$k\geq 0$的Raviart-Thomas空间逼近拟应力张量，并用次数$\le k+1$的连续分片多项式逼近速度场。我们获得了相应Galerkin格式的稳定性、收敛性及先验误差估计。此外，针对任意多边形与多面体区域上的该问题，我们推导出两种可靠且高效的残差型后验误差指示子。所提出指示子的可靠性主要依赖于相关形式的均匀椭圆性、数据适定性假设、稳定的Helmholtz分解以及Clément和Raviart-Thomas算子的局部逼近性质。而逆不等式、基于泡函数的局部化技术及前人工作的已知结果，则是效率估计的主要工具。最后，我们报告了若干数值算例，以验证混合有限元方法的性能、确认理论收敛速率与指示子性质，并展示相关自适应算法的行为。特别地，考虑了含裂缝网络的二维多孔介质流动情形。