  We present three high-order Nystrom discretization strategies of various
boundary integral equation formulations of the impenetrable time-harmonic
Navier equations in two dimensions. One class of such formulations is based on
the four classical Boundary Integral Operators (BIOs) associated with the
Green's function of the Navier operator. We consider two types of Nystrom
discretizations of these operators, one that relies on Kussmaul-Martensen
logarithmic splittings and the other on Alpert quadratures. In addition, we
consider an alternative formulation of Navier scattering problems based on
Helmholtz decompositions of the elastic fields, which can be solved via a
system of boundary integral equations that feature integral operators
associated with the Helmholtz equation. Owing to the fact that some of the BIOs
that are featured in those formulations are non-standard, we use Quadrature by
Expansion (QBX) methods for their high order Nystrom discretization.
Alternatively, we use Maue integration by parts techniques to recast those
non-standard operators in terms of single and double layer Helmholtz BIOs whose
Nystrom discretizations is amenable to the Kussmaul-Martensen methodology. We
present a variety of numerical results concerning the high order accuracy that
our Nystrom discretization elastic scattering solvers achieve for both smooth
and Lipschitz boundaries. We also present extensive comparisons regarding the
iterative behavior of solvers based on different integral equations in the high
frequency regime. Finally, we illustrate how some of the Nystrom
discretizations we considered can be incorporated seamlessly into the
Convolution Quadrature (CQ) methodology to deliver high-order solutions of the
time domain elastic scattering problems.
