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We introduce and analyze various Regularized Combined Field IntegralEquations (CFIER) formulations of time-harmonic Navier equations in media withpiece-wise constant material properties. These formulations can be derivedsystematically starting from suitable coercive approximations ofDirichlet-to-Neumann operators (DtN), and we present a periodicpseudodifferential calculus framework within which the well posedness of CIERformulations can be established. We also use the DtN approximations to deriveand analyze Optimized Schwarz (OS) methods for the solution of elastodynamicstransmission problems. The pseudodifferential calculus we develop in this paperrelies on careful singularity splittings of the kernels of Navier boundaryintegral operators which is also the basis of high-order Nystr\"om quadraturesfor their discretizations. Based on these high-order discretizations weinvestigate the rate of convergence of iterative solvers applied to CFIER andOS formulations of scattering and transmission problems. We present a varietyof numerical results that illustrate that the CFIER methodology leads toimportant computational savings over the classical CFIE one, whenever iterativesolvers are used for the solution of the ensuing discretized boundary integralequations. Finally, we show that the OS methods are competitive in thehigh-frequency high-contrast regime.

Boundary integral equation methods for the solution of scattering and transmission 2D elastodynamic problems