  Helmholtz decompositions of elastic fields is a common approach for the
solution of Navier scattering problems. Used in the context of Boundary
Integral Equations (BIE), this approach affords solutions of Navier problems
via the simpler Helmholtz boundary integral operators (BIOs). Approximations of
Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing
combined field strategy to deliver BIE formulations of the second kind for the
solution of Navier scattering problems in two dimensions with Dirichlet
boundary conditions, at least in the case of smooth boundaries. Unlike the case
of scattering and transmission Helmholtz problems, the approximations of the
DtN maps we use in the Helmholtz decomposition BIE in the Navier case require
incorporation of lower order terms in their pseudodifferential asymptotic
expansions. The presence of these lower order terms in the Navier regularized
BIE formulations complicates the stability analysis of their Nystr\"om
discretizations in the framework of global trigonometric interpolation and the
Kussmaul-Martensen kernel singularity splitting strategy. The main difficulty
stems from compositions of pseudodifferential operators of opposite orders,
whose Nystr\"om discretization must be performed with care via
pseudodifferential expansions beyond the principal symbol. The error analysis
is significantly simpler in the case of arclength boundary parametrizations and
considerably more involved in the case of general smooth parametrizations which
are typically encountered in the description of one dimensional closed curves.
