Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities

TL;DR

This paper investigates transmission operators for the non-overlapping Schwarz method tailored to Helmholtz problems in cavities, emphasizing back-propagating waves and demonstrating a 46% reduction in iteration count in a 3D acoustic noise simulation.

Contribution

It introduces new transmission operators that account for back-propagating waves in cavity problems and compares their performance with traditional operators.

Findings

Operators considering back-propagating waves improve convergence.

Optimized operators reduce iteration count by 46%.

Study includes rectangular cavities and deviations from ideal geometry.

Abstract

In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on the case of rectangular cavities, as the optimal (non-local) transmission operator can be easily determined. Nonetheless, deviations from this ideal geometry are considered as well. In particular, computations of the acoustic noise in a three-dimensional model of the helium vessel of a beamline cryostat with optimized Schwarz schemes are discussed. Those computations show a reduction of 46% in the iteration count, when comparing an operator optimized for cavities with those optimized for unbounded problems.