We investigate a multidimensional transmission problem between viscoelastic system with localized Kelvin-Voigt damping and purely elastic system under different types of geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. It was shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system. So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. For this aim, using frequency domain approach combined with a new multiplier technic, we will establish a polynomial energy decay estimate of type $ t^{-1} $ for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type $ t^{-2/5} $ if the strip is localized near the boundary. Else, it's of type $ t^{-1/3} $. The main novelty in this paper is that the geometric situations covered here are richer and less restrictive than those considered in [
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Elastic-viscoelastic waves interaction models satisfying the assumption (A1)
A model satisfying assumption (A2)
A model satisfying both (A1) and (A2)
A square model with local viscoelastic strip not satisfying any geometry cited before
A model satisfying
A model not satisfying