## Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part III

Gabriele Ciaramella and Martin J. Gander

### Abstract

In the ddCOSMO solvation model for the numerical simulation of molecules (chains of atoms), the unusual observation was made that the associated Schwarz domain-decomposition method converges independently of the number of subdomains (atoms) and this without coarse correction, i.e., the one-level Schwarz method is scalable. We analyzed this unusual property for the simplified case of a rectangular molecule and square subdomains using Fourier analysis, leading to robust convergence estimates in the $L^2$-norm and later also for chains of subdomains represented by disks using maximum principle arguments, leading to robust convergence estimates in $L^{\infty}$. A convergence analysis in the more natural $H^1$-setting proving convergence independently of the number of subdomains was, however, missing. We close this gap in this paper using tools from the theory of alternating projection methods and estimates introduced by P.-L. Lions for the study of domain decomposition methods. We prove that robust convergence independently of the number of subdomains is possible also in $H^1$ and show furthermore that even for certain two-dimensional domains with holes, Schwarz methods can be scalable without coarse-space corrections. As a by-product, we review some of the results of P.-L. Lions [On the Schwarz alternating method. I, in Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp. 1–42] and in some cases provide simpler proofs.

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### Key words

domain decomposition methods, Schwarz methods, chain of subdomains, elliptic PDE, Laplace equation, COSMO solvation model

### AMS subject classifications

65N55, 65F10, 65N22, 70-08, 35J05, 35J57