# Multitrace formulations and Domain Decomposition Methods for the solution of Helmholtz transmission problems for bounded composite scatterers

## TL;DR

This paper develops and analyzes multitrace formulations and domain decomposition methods for efficiently solving Helmholtz transmission problems in composite scatterers, demonstrating improved iterative solver performance.

## Contribution

It introduces a novel DDM approach with generalized Robin boundary conditions using Fourier multiplier approximations, enhancing solver efficiency for Helmholtz problems.

## Key findings

- Hierarchical elimination improves Krylov solver efficiency
- Generalized Robin conditions reduce iteration counts
- Method scales mildly with frequency and subdomain number

## Abstract

We present Nystr\"om discretizations of multitrace formulations and non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for bounded composite scatterers with piecewise constant material properties. We investigate the performance of DDM with both classical Robin and generalized Robin boundary conditions. The generalized Robin boundary conditions incorporate square root Fourier multiplier approximations of Dirichlet to Neumann operators. While the classical version of DDM is not particularly well suited for Krylov subspace iterative solvers, we show that the associated DDM linear system can be efficiently solved by hierarchical elimination via Schur complements of the Robin data. We show through numerical examples that the latter version of DDM gives rise to small numbers of Krylov subspace iterations that depend mildly on the frequency and number of subdomains.