A numerical algorithm based on probing to find optimized transmission conditions

Optimized Schwarz Methods (OSMs) are very versatile: they can be used with or without overlap, converge faster compared to other domain decomposition methods [5], are among the fastest solvers for wave problems [10], and can be robust for heterogeneous problems [7]. This is due to their general transmission conditions, optimized for the problem at hand. Over the last two decades such conditions have been derived for many Partial Differential Equations (PDEs), see [7] for a review. Optimized transmission conditions can be obtained by diagonalizing the OSM iteration using a Fourier transform for two subdomains with a straight interface. This works surprisingly well, but there are important cases where the Fourier approach fails: geometries with curved interfaces (there are studies for specific geometries, e.g. [11, 9, 8]), and heterogeneous couplings when the two coupled problems are quite different in terms of eigenvectors of the local Steklov-Poincaré operators [6]. There is therefore a great need for numerical routines which allow one to get cheaply optimized transmission conditions, which furthermore could then lead to OSM black-box solvers. Our goal is to present one such procedure. Let us consider the simple case of a two nonoverlapping subdomain decomposition, that is Ω = Ω1 ∪ Ω2, Ω1 ∩ Ω2 = ∅, Γ := Ω1 ∩ Ω2, and a generic second order linear PDE L(u) = f , in Ω, u = 0 on ∂Ω. (1)