In this work, we consider wave propagation in harmonic regime modelled by the Helmholtz equation. We discuss a solution strategy based on the Generalized Optimized Schwarz Method (GOSM) introduced in [Claeys, 2021], which leads to a reformulation of the problem as an equation posed on the skeleton of the subdomain partition. In this approach, the wave equation is imposed separately in each subdomain, and transmission conditions coupling subdomains are imposed by means of a so-called exchange operator that can be non-local.

In the present talk, we are specifically interested in the Helmholtz equation posed in a complex configuration that may involve physical boundaries (e.g. Neumann or Dirichlet boundary conditions) and boundary integral equation contributions (FEM-BEM coupling), a situation that allows resonance phenomena. We shall describe how to adapt GOSM to this extended setting, what are the theoretical outcomes of this approach.

Joint work with A. Boisneault, M. Bonazzoli and P. Marchand.

References

• X.Claeys. "Non-local variant of the optimised Schwarz method for arbitrary non-overlapping subdomain partitions". ESAIM: Math. Model. Numer. Anal. (2021).
• X.Claeys, "Non-local optimized Schwarz method for the Helmholtz equation with physical boundaries", SIAM J. Math. Anal., vol.55, no.6, pp.7490-7512, (2023).
• A.Boisneault, M.Bonazzoli, X.Claeys, P.Marchand, "Discrete FEM-BEM coupling with the Generalized Optimized Schwarz Method", submitted, preprint arXiv:2601.16817