We consider a mixed variational formulation recently proposed for the coupling of the Brinkman--Forchheimer and Darcy equations and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D version of the associated conforming mixed finite element scheme. For the reliability analysis, due to the nonlinear nature of the problem, we make use of the inf-sup condition and the strong monotonicity of the operators involved, along with a stable Helmholtz decomposition in Hilbert spaces and local approximation properties of the Raviart--Thomas and Cl\'ement interpolants. On the other hand, inverse inequalities, the localization technique through bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms are reported. In particular, the case of flow through a heterogeneous porous medium is considered.

翻译：我们研究了近期提出的Brinkman-Forchheimer方程与Darcy方程耦合的混合变分形式，并针对其二维相容混合有限元格式首次建立了可靠且高效的残差型后验误差估计器。在可靠性分析中，由于问题的非线性特性，我们利用了算子的inf-sup条件与强单调性，并结合Hilbert空间中稳定的Helmholtz分解、Raviart-Thomas插值与Clément插值的局部逼近性质。另一方面，逆不等式、通过气泡函数实现的局部化技术以及已有文献中的结论是获得效率估计的主要工具。最后，本文通过若干数值算例验证了估计器的理论性质，并展示了相关自适应算法的性能表现，特别考虑了流体在非均质多孔介质中流动的案例。