Analysis of a Schwarz-Fourier domain decomposition method

TL;DR

This paper analyzes a Fourier-based Schwarz domain decomposition method for Laplace equations, providing convergence insights and bounds relevant for computational chemistry applications.

Contribution

It introduces a new convergence analysis for an inexact Schwarz method using Fourier projections, with novel maximum principle bounds.

Findings

Convergence of the Schwarz-Fourier method is established.

Derived new maximum principle variants and contraction bounds.

Results are applicable to computational chemistry solvers like ddCOSMO.

Abstract

The Schwarz domain decomposition method can be used for approximately solving a Laplace equation on a domain formed by the union of two overlapping discs. We consider an inexact variant of this method in which the subproblems on the discs are solved approximately using the projection on a Fourier subspace of the $L^2$ space on the boundary. This model problem is relevant for better understanding of the ddCOSMO solver that is used in computational chemistry. We analyze convergence properties of this Schwarz-Fourier domain decomposition method. The analysis is based on maximum principle arguments. We derive a new variant of the maximum principle and contraction number bounds in the maximum norm.