Section: New Results

Integral equation based optimized Schwarz method for electromagnetics

The optimized Schwarz method (OSM) is recognised as one of the most efficient domain decomposition strategies without overlap for the solution to wave propagation problems in harmonic regime. For the Helmholtz equation, this approach originated from the seminal work of Després, and led to the development of an abundent literature offering more elaborated but more efficient transmission conditions. Most contributions focus on transmission conditions based on local operators.

In recent years, F. Collino, P. Joly and M. Lecouvez introduced non-local transmission conditions that can drastically improve the convergence rate of OSM. The performance of this strategy seems to remain robust at high frequency. Such an approach was proposed only for the Helmholtz equation, and has still not been adapted to electromagnetics.

In this work we investigated such an approach for Maxwell's equations in a simple spherical geometry that allows explicit calculus by means of separation of variables. The transmission condition that we propose involves a non-local operator that is a dissipative counterpart of the so-called Electric Field integral operator (EFIE) which is a classical object in electromagnetic potential theory. We show that the iterative solver associated to our strategy converges at an exponential rate.