  This paper presents novel methodologies for the numerical simulation of
scattering of elastic waves by both closed and open surfaces in
three-dimensional space. The proposed approach utilizes new integral
formulations as well as an extension to the elastic context of the efficient
high-order singular-integration methods~\cite{BG18} introduced recently for the
acoustic case. In order to obtain formulations leading to iterative solvers
(GMRES) which converge in small numbers of iterations we investigate,
theoretically and computationally, the character of the spectra of various
operators associated with the elastic-wave Calder\'on relation---including some
of their possible compositions and combinations. In particular, by relying on
the fact that the eigenvalues of the composite operator $NS$ are bounded away
from zero and infinity, new uniquely-solvable, low-GMRES-iteration integral
formulation for the closed-surface case are presented. The introduction of
corresponding low-GMRES-iteration equations for the open-surface equations
additionally requires, for both spectral quality as well as accuracy and
efficiency, use of weighted versions of the classical integral operators to
match the singularity of the unknown density at edges. Several numerical
examples demonstrate the accuracy and efficiency of the proposed methodology.
