This paper introduces a novel approach to analyzing overlapping Schwarz methods for N\'{e}d\'{e}lec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the topology of the domain. Enhanced estimates for the condition numbers of the preconditioned linear systems are derived, dependent linearly on the relative overlap between the overlapping subdomains. Furthermore, we present the numerical experiments which support our theoretical results.

翻译：本文提出了一种分析Nédélec和Raviart-Thomas向量场问题重叠Schwarz方法的新思路。该理论基于对向量场在域拓扑结构下具有鲁棒性的新的正则稳定分解。我们推导出了预处理线性系统条件数改进的估计，该估计依赖于重叠子域间相对重叠度的线性关系。此外，我们提供的数值实验验证了理论结果的正确性。