  In this work, we focus on the Neumann-Neumann method (NNM), which is one of
the most popular non-overlapping domain decomposition methods. Even though the
NNM is widely used and proves itself very efficient when applied to discrete
problems in practical applications, it is in general not well defined at the
continuous level when the geometric decomposition involves cross-points. Our
goals are to investigate this well-posedness issue and to provide a complete
analysis of the method at the continuous level, when applied to a simple
elliptic problem on a configuration involving one cross-point. More
specifically, we prove that the algorithm generates solutions that are singular
near the cross-points. We also exhibit the type of singularity introduced by
the method, and show how it propagates through the iterations. Then, based on
this analysis, we design a new set of transmission conditions that makes the
new NNM geometrically convergent for this simple configuration. Finally, we
illustrate our results with numerical experiments.
