A unified theory of non-overlapping Robin-Schwarz methods -- continuous and discrete, including cross points

Abstract

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.