In two-dimensional Lipschitz domains, we analyze 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 prove the existence and uniqueness of a solution. We propose a finite element method and obtain a quasi-best approximation result in the energy norm \emph{\`a la C\'ea} under the assumption that $\Omega$ is convex. We also develop an a posteriori error estimator and study its reliability and efficiency properties. Finally, we develop an adaptive method that yields optimal experimental convergence rates for the numerical examples we perform.

翻译：在二维Lipschitz域上，我们分析了加权空间 $\mathbf{H}_0^1(\omega,\Omega) \times L^2(\omega,\Omega)/\mathbb{R}$ 中的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方法，并在 $\Omega$ 为凸域的假设下，获得了能量范数下的拟最佳逼近结果（类似Céa引理）。我们还开发了一种后验误差估计器，并研究了其可靠性和效率性质。最后，我们设计了一种自适应方法，在数值算例中实现了最优实验收敛速度。