A posteriori estimate and asymptotic partial domain decomposition

The method of asymptotic partial decomposition of a domain aims at replacing a 3D or 2D problem by a hybrid problem 3D???1D; or 2D???1D, where the dimension of the problem decreases in part of the domain. The location of the junction between the heterogeneous problems is asymptotically estimated in certain circumstances, but for numerical simulations it is important to be able to determine the location of the junction accurately. In this article, by reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. Some numerical results are presented for a toy problem.