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.

翻译：本文聚焦于Neumann-Neumann方法（NNM），这是最流行的非重叠区域分解方法之一。尽管NNM被广泛使用并在实际应用的离散问题中展现出高效性，但当几何分解涉及交叉点时，该方法在连续层面通常不具有良好的定义。我们的目标是研究这一适定性问题，并在包含一个交叉点的简单椭圆问题配置下，对该方法在连续层面进行完整分析。具体而言，我们证明了算法会在交叉点附近产生奇异解，揭示了该方法引入的奇异性类型，并展示了该奇异性如何随迭代传播。基于这一分析，我们设计了一组新的传输条件，使得新NNM在此简单配置下实现几何收敛。最后，我们通过数值实验验证了研究结果。