We consider the polynomial stability for an abstract system of the type $u_{tt} + Lu + B u_{t} = 0,$ where $L$ is a self-adjoint operator on a Hilbert space and operator $B$ represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator $L$ and non-degeneration of operator $B$.  As an application of our abstract results, we give the decay rate of the energy for a wave equation with local viscous or viscoelastic damping.