Convergence of classical optimized non-overlapping Schwarz method for Helmholtz problems in closed domains

TL;DR

This paper analyzes the convergence of advanced optimized Schwarz methods for Helmholtz problems in closed cavity domains, focusing on the effects of back-propagating waves and evaluating various transmission conditions.

Contribution

It provides a detailed analysis of convergence behavior for optimized Schwarz methods in closed domains and compares different transmission conditions' effectiveness.

Findings

Back-propagating waves significantly affect convergence.

Optimized transmission conditions improve solution accuracy.

Performance varies with different transmission condition orders.

Abstract

In this paper we discuss the convergence of state-of-the-art optimized Schwarz transmission conditions for Helmholtz problems defined on closed domains (i.e. setups which do not exhibit an outgoing wave condition), as commonly encountered when modeling cavities. In particular, the impact of back-propagating waves on the Dirichlet-to-Neumann map is analyzed. Afterwards, the performance of the well-established optimized 0th-order, evanescent modes damping, optimized 2nd-order and Pad\'e-localized square-root transmission conditions is discussed.