  Non-overlapping Schwarz methods with generalized Robin transmission
conditions were originally introduced by B. Despr\'es for time-harmonic wave
propagation problems and have largely developed over the past thirty years. The
aim of the paper is to provide both a review of the available formulations and
methods as well as a consistent theory applicable to more general cases than
studied until to date. An abstract variational framework is provided
reformulating the original problem by the well-known form involving a
scattering operator and an interface exchange operator, and the equivalence
between the formulations is discussed thoroughly. The framework applies to a
series of wave propagation problems throughout the de Rham complex, such as the
scalar Helmholtz equation, Maxwell's equations, a dual formulation of the
Helmholtz equation in H(div), as well as any conforming finite element
discretization thereof, and it applies also to coercive problems. Three
convergence results are shown. The first one (using compactness) and the second
one (based on absorbtion) generalize Despr\'es' early findings and apply as
well to the FETI-2LM formulation. The third result, oriented on the work by
Collino, Ghanemi, and Joly, establishes a convergence rate and covers cases
with cross points, while not requiring any regularity of the solution. The key
ingredient is a global interface exchange operator, proposed originally by X.
Claeys and further developed by Claeys and Parolin, here worked out in full
generality. The third type of convergence theory is applicable at the discrete
level as well, where the exchange operator is allowed to be even local. The
resulting scheme can be viewed as a generalization of the 2-Lagrange-multiplier
method introduced by S. Loisel, and connections are drawn to another technique
proposed by Gander and Santugini.
