SIMAI 2025

FEM-BEM coupling with a substructured domain decomposition method

  • Boisneault, Antonin (POEMS, CNRS, Inria, ENSTA)
  • Bonazzoli, Marcella (Inria, UMA, ENSTA)
  • Claeys, Xavier (POEMS, CNRS, Inria, ENSTA)
  • Marchand, Pierre (POEMS, CNRS, Inria, ENSTA)

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In this talk we are interested in simulating time-harmonic acoustic wave propagation, modeled by the Helmholtz equation, in a complex medium composed of a bounded heterogeneous part and an unbounded homogeneous part. The bounded part is decomposed into non overlapping subdomains, with cross-points allowed, i.e. points where three or more subdomains are adjacent. Our solution strategy is based on the Generalized Optimized Schwarz Method (GOSM) introduced in [Claeys 2021, Claeys 2023], which leads to a substructured reformulation of the problem as an equation posed on the skeleton of the domain partition. In this approach, the wave equation is imposed separately in each subdomain, and transmission conditions coupling the subdomains are imposed by means of a so-called exchange operator that can be non-local. We show that the GOSM framework in [Claeys 2023] can be extended to cover FEM-BEM coupling approaches, such as the symmetric Costabel coupling [Costabel 1987], which couple boundary integral equations for the homogeneous part with volume variational formulations for the heterogeneous part. We are able to prove that the GOSM for the Costabel FEM-BEM coupling is geometrically convergent for iterative procedures such as GMRES or Richardson methods. We present extensive numerical experiments to illustrate the method convergence, and discuss the impact of the choice of (non-local) transmission operators.