Title: Analysis of the parallel Schwarz method for growing chains of fixed-sized
subdomains
Abstract:
A new class of Schwarz methods was recently presented in the literature for the
solution of solvation models, where the electrostatic energy contribution to the
solvation energy can be computed by solving a system of elliptic partial differen-
tial equations [1,2]. Numerical simulations have shown an unusual convergence
behaviour of Schwarz methods for the solution of this problem, where each atom
corresponds to a subdomain: the convergence of the Schwarz methods is inde-
pendent of the number of atoms [1], even though there is no coarse grid correc-
tion. Despite the successful implementation of Schwarz methods for this solvation
model, a rigorous analysis for this unusual convergence behaviour is required, since
no theoretical results are given in the corresponding literature.
In this talk, we analyze the behavior of the Schwarz method for the solution of a
chain of atoms and show that its convergence does not depend on the number of
atoms (subdomains). We use two different techniques to prove this result. The
first technique is based on a Fourier expansion of the error and the analysis of
transfer matrices constructed for an approximate model. The second one consists
in an application of the maximum-principle and allows us to analyze very general
geometries.
[1] Cancès et al., Domain decomposition for implicit solvation models, J. of Chem.
P. (2013).
[2] Lipparini et al., Fast Domain Decomposition Algorithm for Continuum Solva-
tion Models, J. Chem. Theory Comput. (2013).
