  Consider the elastic scattering of a time-harmonic wave by multiple well
separated rigid particles in two dimensions. To avoid using the complex Green's
tensor of the elastic wave equation, we utilize the Helmholtz decomposition to
convert the boundary value problem of the elastic wave equation into a coupled
boundary value problem of Helmholtz equations. Based on single, double, and
combined layer potentials with the simpler Green's function of the Helmholtz
equation, we present three different boundary integral equations for the
coupled boundary value problem. The well-posedness of the new integral
equations are established. Computationally, a scattering matrix based method is
proposed to evaluate the elastic wave for arbitrarily shaped particles. The
method uses the local expansion for the incident wave and the multipole
expansion for the scattered wave. The linear system of algebraic equations is
solved by GMRES with fast multipole method (FMM) acceleration. Numerical
results show that the method is fast and highly accurate for solving the
elastic scattering problem with multiple particles.
