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We study the Schwarz overlapping domain decomposition method applied to thePoisson problem on a special family of domains, which by construction consistof a union of a large number of fixed-size subdomains. These domains aremotivated by applications in computational chemistry where the subdomainsconsist of van der Waals balls. As is usual in the theory of domaindecomposition methods, the rate of convergence of the Schwarz method is relatedto a stable subspace decomposition. We derive such a stable decomposition forthis family of domains and analyze how the stability "constant" depends onrelevant geometric properties of the domain. For this, we introduce newdescriptors 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 ofthe Schwarz method depends on specific local geometry descriptors and on oneglobal geometry descriptor. The analysis also naturally provides lower boundsin terms of the descriptors for the smallest eigenvalue of the Laplaceeigenvalue problem for this family of domains.

Analysis of the Schwarz domain decomposition method for the conductor-like screening continuum model