SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 769-796, January 2021.
We study the Schwarz overlapping domain decomposition method applied to the Poisson problem on a special family of domains, which by construction consist of a union of a large number of fixed-size subdomains. These domains are motivated by applications in computational chemistry where the subdomains consist of van der Waals balls. As is usual in the theory of domain decomposition methods, the rate of convergence of the Schwarz method is related to a stable subspace decomposition. We derive such a stable decomposition for this family of domains and analyze how the stability “constant" depends on relevant geometric properties of the domain. For this, we introduce new descriptors that are used to formalize the geometry for the family of domains. We show how, for an increasing number of subdomains, the rate of convergence of the Schwarz method depends on specific local geometry descriptors and on one global geometry descriptor. The analysis also naturally provides lower bounds in terms of the descriptors for the smallest eigenvalue of the Laplace eigenvalue problem for this family of domains.

中文翻译：

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类导体屏蔽连续体模型的Schwarz域分解方法分析

SIAM数值分析杂志，第59卷，第2期，第769-796页，2021年1月。
我们研究了应用于特殊域族的Poisson问题的Schwarz重叠域分解方法，该方法通过构造由大量固定大小的子域的并集构成。这些域是由计算化学中的应用激发的，其中子域由范德华斯球组成。与域分解方法理论中的通常情况一样，Schwarz方法的收敛速度与稳定的子空间分解有关。我们对该域族导出了这样的稳定​​分解，并分析了稳定性“常数”如何依赖于域的相关几何特性，为此，我们引入了新的描述符，用于对域族的几何形式进行形式化。对于越来越多的子域，Schwarz方法的收敛速度取决于特定的局部几何形状描述符和一个全局几何形状描述符。该分析自然也就该族域的拉普拉斯特征值问题的最小特征值的描述符提供了下界。