We propose a numerical method to approximate the scattering amplitudes for the elasticity system with a non-constant matrix potential in dimensions $d=2$ and $3$. This requires to approximate first the scattering field, for some incident waves, which can be written as the solution of a suitable Lippmann-Schwinger equation. In this work we adapt the method introduced by G. Vainikko in cite{V} to solve such equations when considering the Lame operator. Convergence is proved for sufficiently smooth potentials. Implementation details and numerical examples are also given.
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