  In this paper we present a full discretization of the layer potentials and
boundary integral operators for the elastic wave equation on a parametrizable
smooth closed curve in the plane. The method can be understood as a
non-conforming Petrov-Galerkin discretization, with a very precise choice of
testing functions by symmetrically combining elements on two staggered grids,
and using a look-around quadrature formula. Unlike in the acoustic counterpart
of this work, the kernel of the elastic double layer operator includes a
periodic Hilbert transform that requires a particular choice of the mixing
parameters. We give mathematical justification of this fact. Finally, we test
the method on some frequency domain and time domain problems, and demonstrate
its applicability on smooth open arcs.
