Non-local optimized Schwarz method with physical boundaries

TL;DR

This paper extends non-local optimized Schwarz methods to bounded Helmholtz problems with physical boundaries, providing a theoretical framework that includes resonance phenomena and establishes coercivity under certain conditions.

Contribution

It introduces a new analysis of non-local optimized Schwarz methods for Helmholtz equations with physical boundaries, including resonance considerations and explicit coercivity bounds.

Findings

The skeleton formulation is coercive when the problem is uniquely solvable.

Explicit bounds for the coercivity constant are derived.

Resonance phenomena are incorporated into the analysis.

Abstract

We extend the theoretical framework of non-local optimized Schwarz methods as introduced in [Claeys,2021], considering an Helmholtz equation posed in a bounded cavity supplemented with a variety of conditions modeling material boundaries. The problem is reformulated equivalently as an equation posed on the skeleton of a non-overlapping partition of the computational domain, involving an operator of the form "identity + contraction". The analysis covers the possibility of resonance phenomena where the Helmholtz problem is not uniquely solvable. In case of unique solvability, the skeleton formulation is proved coercive, and an explicit bound for the coercivity constant is provided in terms of the inf-sup constant of the primary Helmholtz boundary value problem.