  We study the Landau damping of ferromagnetic magnons in Fe, Co, and Ni as the
dimensionality of the system is reduced from three to two. We resort to the
\textit{ab initio} linear response time dependent density functional theory in
the adiabatic local spin density approximation. The numerical scheme is based
on the Korringa-Kohn-Rostoker Green's function method. The key points of the
theoretical approach and the implementation are discussed. We investigate the
transition metals in three different forms: bulk phases, free-standing thin
films and thin films supported on a nonmagnetic substrate. We demonstrate that
the dimensionality trends in Fe and Ni are opposite: in Fe the transition from
bulk bcc crystal to Fe/Cu(100) film reduces the damping whereas in Ni/Cu(100)
film the attenuation increases compared to bulk fcc Ni. In Co, the strength of
the damping depends relatively weakly on the sample dimensionality. We explain
the difference in the trends on the basis of the underlying electronic
structure. The influence of the substrate on the spin-wave damping is analyzed
by employing Landau maps representing wave-vector resolved spectral density of
the Stoner excitations.
