Multi-domain FEM-BEM coupling with several impenetrable obstacles

In a recent paper we have analyzed a new formulation of the coupling of finite and boundary element methods (FEM-BEM) for Helmholtz problems, involving several heterogeneous bounded subdomains and one homogeneous unbounded subdomain.

This formulation, called Generalized Optimized Schwarz Method (GOSM), is substructured, that is, its unknowns are associated with the subdomains interfaces.

To derive the GOSM, the first step was to prove that a solution to the Helmholtz problem satisfies a specific multi-domain variational formulation, which involves one operator for each subdomain, each operator being independent of the others.

In the present contribution, we design a variational formulation of that type for a more general geometrical and material configuration: several heterogeneous bounded subdomains, impenetrable obstacles and homogeneous subdomains are allowed.

Note that, like in our recent paper, we assume that only one subdomain is unbounded, and that its boundary is bounded.

The domain partition can have cross-points, that is, points where at least three subdomains are adjacent.

We also prove that a solution to the initial Helmholtz problem can be recovered from a solution to the multi-domain variational formulation.

This shows that the GOSM is a general and flexible framework to model acoustic wave propagation, as it can handle both multi-domain FEM-BEM coupling and (weakly imposed) boundary conditions on several obstacles.