Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

TL;DR

This paper analyzes the convergence of parallel Schwarz algorithms for wave problems, demonstrating scalability under certain conditions and leveraging block Toeplitz matrix structures for theoretical insights.

Contribution

It extends convergence results for Schwarz methods with impedance conditions to continuous wave problems using block Toeplitz matrices, showing scalability without coarse spaces.

Findings

Convergence is achieved for wave problems with impedance conditions under specific assumptions.

The global iteration matrix exhibits a block Toeplitz structure with a spectrum similar to Hermitian matrices.

Numerical experiments confirm the theoretical convergence properties.

Abstract

In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions applied to the one-dimensional and two-dimensional Helmholtz and Maxwell's equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm applied to the equations with absorption, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that can typically be encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments.