An a posteriori error estimate for a 0D/2D coupled model

Abstract: 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}