## Section: New Results

### Multi-subdomain integral equations

In the context of boundary integral equations adapted to wave scattering in piecewise constant media in harmonic regime, we also made significant progress in the study of the single trace boundary integral formulation (STF) of the second kind originally introduced in [17]. This work was achieved in collaboration with Ralf Hiptmair and Elke Spindler (ETH Zürich). First of all, we proposed a version of this formulation for the solution to Maxwell's equations whereas, so far, it had been studied only in the context of scalar wave scattering (Helmholtz equation). In this direction, we conducted numerical experiments which confirmed the attractive properties of the matrices obtained when discretising such formulations (good accuracy, and good conditionning independent of discretisation parameters). For Maxwell's equations, we also established elementary theoretical results of STF 2nd kind such as Fredholmness of the corresponding integral operator.

So far, second kind STF had been stuudied for wave scatering problems where material contrasts only enter in the compact part of the partial differential operator, which is harmless regarding the Fredholmness of the corresponding boundary integral operator. Thus, in [19], we investigated the case where material contrasts come into play in the principal part of the operator, considering a pure diffusion-transmission problem. In this case, we have been able to establish well-posedness (hence Fredholmness). A rather naive approach leads to choose Sobolev spaces of fractionnal order (half-integer) as main functional setting for this formulation. We showed that this formulation can be extended so as to make sense in the space of square integrable trace functions. This is much more handy a functional setting that allows in particular discontinuous Galerkin discretisations of the corresponding boundary integral equations.