  We study for the first time Schwarz domain decomposition methods for the
solution of the Navier equations modeling the propagation of elastic waves.
These equations in the time harmonic regime are difficult to solve by iterative
methods, even more so than the Helmholtz equation. We first prove that the
classical Schwarz method is not convergent when applied to the Navier
equations, and can thus not be used as an iterative solver, only as a
preconditioner for a Krylov method. We then introduce more natural transmission
conditions between the subdomains, and show that if the overlap is not too
small, this new Schwarz method is convergent. We illustrate our results with
numerical experiments, both for situations covered by our technical two
subdomain analysis, and situations that go far beyond, including many
subdomains, cross points, heterogeneous materials in a transmission problem,
and Krylov acceleration. Our numerical results show that the Schwarz method
with adapted transmission conditions leads systematically to a better solver
for the Navier equations than the classical Schwarz method.
