Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part III. ETNA - Electronic Transactions on Numerical Analysis

Résumé 0

In the ddCOSMO solvation model for the numerical simulation ofmolecules (chains of atoms), the unusual observation was made thatthe associated Schwarz domain-decomposition method converges independentlyof the number of subdomains (atoms) and this without coarsecorrection, i.e., the one-level Schwarz method is scalable.We analyzed this unusual property for the simplified caseof a rectangular molecule and square subdomains using Fourier analysis,leading to robust convergence estimates in the $L^2$-norm and lateralso for chains of subdomains represented by disks using maximumprinciple 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 ofsubdomains was, however, missing. We close this gap in this paperusing tools from the theory of alternating projection methodsand estimates introduced by P.-L. Lions for the study of domaindecomposition methods. We prove that robust convergenceindependently of the number of subdomains is possible also in $H^1$and show furthermore that even for certain two-dimensional domainswith 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 DomainDecomposition Methods for Partial Differential Equations, SIAM,Philadelphia, 1988, pp. 1-42]and in some cases provide simpler proofs.