This work is motivated by the need of efficient numerical simulations of gas flows in the serpentine channels used in proton-exchange membrane fuel cells. In particular, we consider the Poisson problem in a 2D domain composed of several long straight rectangular sections and of several bends corners. In order to speed up the resolution, we propose a 0D model in the rectangular parts of the channel and a Finite Element resolution in the bends. To find a good compromise between precision and time consuming, the challenge is double: how to choose a suitable position of the interface between the 0D and the 2D models and how to control the discretization error in the bends. We shall present an \textit{a posteriori} error estimator based on an equilibrated flux reconstruction in the subdomains where the Finite Element method is applied. The estimates give a global upper bound on the error measured in the energy norm of the difference between the exact and approximate solutions on the whole domain. They are guaranteed, meaning that they feature no undetermined constants. (global) Lower bounds for the error are also derived. An adaptive algorithm is proposed to use smartly the estimator for aforementioned double challenge. A numerical validation of the estimator and the algorithm completes the work. \end{abstract}

翻译：本研究旨在满足质子交换膜燃料电池蛇形通道内气体流动高效数值模拟的需求。具体而言，我们考虑由若干长直矩形段和若干弯道构成的二维区域上的泊松问题。为加快求解速度，在通道的矩形部分提出0D模型，并在弯道处采用有限元求解。为在计算精度与耗时之间取得良好平衡，面临双重挑战：如何合理选择0D与2D模型之间的界面位置，以及如何控制弯道处的离散误差。我们将提出一种基于均衡通量重构的后验误差估计器，该估计器适用于应用有限元方法的子区域。该估计给出了整个区域上精确解与近似解之差的能量范数度量的全局误差上界，且具有保证性，即不包含任何未定常数。同时推导了（全局）误差下界。针对上述双重挑战，提出一种自适应算法以智能利用该估计器。数值验证结果证实了估计器及算法的有效性。