On the Relation Between Optimized Schwarz Methods and Source Transfer

Optimized Schwarz methods (OS) use Robin or higher order transmission conditions instead of the classical Dirichlet ones. An optimal Schwarz method for a general second-order elliptic problem and a decomposition into strips was presented in [13]. Here optimality means that the method converges in a finite number of steps, and this was achieved by replacing in the transmission conditions the higher order operator by the subdomain exterior Dirichlet-to-Neumann (DtN) maps. It is even possible to design an optimal Schwarz method that converges in two steps for an arbitrary decomposition and an arbitrary partial differential equation (PDE), see [6], but such algorithms are not practical, because the operators involved are highly non-local. Substantial research was therefore devoted to approximate these optimal transmission conditions, see for example the early reference [11], or the overview [5] which coined the term “optimized Schwarz method”, and references therein. In particular for the Helmholtz equation, Gander et al. [9] presents optimized second-order approximations of the DtN, Toselli [17] (improperly) and Schädle and Zschiedrich [14] (properly) tried for the first time using perfectly matched layers (PML, see [1]) to approximate the DtN in OS.