In Lipschitz two-dimensional domains, we study a Brinkman-Darcy-Forchheimer problem on the weighted spaces $\mathbf{H}_0^1(\omega,\Omega) \times L^2(\omega,\Omega)/\mathbb{R}$, where $\omega$ belongs to the Muckenhoupt class $A_2$. Under a suitable smallness assumption, we establish the existence and uniqueness of a solution. We propose a finite element scheme and obtain a quasi-best approximation result in energy norm \`a la C\'ea under the assumption that $\Omega$ is convex. We also devise an a posteriori error estimator and investigate its reliability and efficiency properties. Finally, we design a simple adaptive strategy that yields optimal experimental rates of convergence for the numerical examples that we perform.

翻译：在二维Lipschitz区域中，我们在加权空间$\mathbf{H}_0^1(\omega,\Omega) \times L^2(\omega,\Omega)/\mathbb{R}$上研究了Brinkman-Darcy-Forchheimer问题，其中$\omega$属于Muckenhoupt类$A_2$。在适当的小性假设下，我们证明了解的存在唯一性。我们提出了一种有限元格式，并在$\Omega$为凸区域的假设下，获得了\`a la C\'ea形式的能量范数拟最优逼近结果。我们还设计了一个后验误差估计子，并研究了其可靠性和效率性质。最后，我们提出了一种简单的自适应策略，该策略在执行的数值算例中实现了最优的实验收敛速率。