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

TL;DR

This paper analyzes the convergence of the Schwarz domain decomposition method applied to the Poisson problem on domains composed of multiple fixed-size subdomains, with applications in computational chemistry.

Contribution

It derives a stable subspace decomposition for domains made of van der Waals balls and relates convergence rates to geometric descriptors of the domain.

Findings

Convergence rate depends on local and global geometric descriptors.

Introduces new geometric descriptors to analyze domain properties.

Provides lower bounds for the Laplace eigenvalues based on geometry.

Abstract

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.