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:在这项工作中,我们的重点是Neumann-Neumann-Neumann方法(NNM),这是最受欢迎的非重叠域分解方法之一。尽管NNM在实际应用中被广泛使用,而且当应用到离散问题时证明非常有效,但一般而言,当几何分解涉及交叉点时,该方法在连续水平上没有很好的定义。我们的目标是调查这个稳妥的储存问题,并在连续水平上对方法进行全面分析,如果该方法应用于一个涉及一个交叉点的配置中简单的椭圆问题。更具体地说,我们证明算法产生的解决方案在交叉点上是独一无二的。我们还展示了该方法引入的单一性类型,并展示了该方法是如何通过迭代传播的。然后,根据这一分析,我们设计了一套新的传输条件,使新的NMM的地理分解为这种简单配置提供了完整的。最后,我们用数字实验来说明我们的结果。</s>