The Acoustic Black Hole (ABH) is an effective passive device for reducing vibrations of mechanical structures without adding mass. In practice, it is realized in plates by manufacturing a pit of quadratic profile giving rise to a gradual decrease in thickness as a function of the radius, according to a power law and by placing a thin layer of viscoelastic material in the central zone. The presence of the ABH in the plate constitutes an inhomogeneity, which give rise to complex scattering mechanisms for bending waves. Characterization of the scattering properties of this particular scatterer constitutes the main objective of this work. The presence of viscoelastic film induces an inhomogeneous damping that is added to that of the material constituting the plate. The study of the dissipation mechanisms is conducted in two steps. The first step consists in developing an analytical model of reference, characterizing the ABH behavior in the absence of viscoelastic coating. The second step is to study numerically with the finite difference method the scattering by the black hole with the viscoelastic film, and extend the results of the reference configuration to realistic configurations which are not accessible analytically.