  We introduce and analyze various Regularized Combined Field Integral
Equations (CFIER) formulations of time-harmonic Navier equations in media with
piece-wise constant material properties. These formulations can be derived
systematically starting from suitable coercive approximations of
Dirichlet-to-Neumann operators (DtN), and we present a periodic
pseudodifferential calculus framework within which the well posedness of CIER
formulations can be established. We also use the DtN approximations to derive
and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics
transmission problems. The pseudodifferential calculus we develop in this paper
relies on careful singularity splittings of the kernels of Navier boundary
integral operators which is also the basis of high-order Nystr\"om quadratures
for their discretizations. Based on these high-order discretizations we
investigate the rate of convergence of iterative solvers applied to CFIER and
OS formulations of scattering and transmission problems. We present a variety
of numerical results that illustrate that the CFIER methodology leads to
important computational savings over the classical CFIE one, whenever iterative
solvers are used for the solution of the ensuing discretized boundary integral
equations. Finally, we show that the OS methods are competitive in the
high-frequency high-contrast regime.
