This paper presents the implementation of the finite element method combined with Dirichlet-to-Neumann (DtN) mapping (derived in terms of an infinite Fourier series) for studying the solvability of an exterior problem arising in linear incomepressible 2D-elasticity. A reliable numerical experiment is also presented showing the accuracy of DtN mapping; only a few Fourier series terms were needed to get a good approximation to the solution. The stable Taylor-Hood element was used for finite element discretisation.