  We present three-body nonrelativistic calculations within the framework of a
potential model for the kaonic cluster ${K^-}pp$ using two completely different
methods: the method of hyperspherical harmonics in the momentum representation
and the method of Faddeev equations in configuration space. To perform a
numerical benchmark, different $NN$ and antikaon-nucleon interactions are
applied. The results of the calculations for the ground state energy for the
${K^-}pp$ system obtained by both methods are in reasonable agreement. Although
the ground state energy is not sensitive to the $NN$ interaction, it shows very
strong dependence on the $\overset{\_}{K}N$ potential. We show that the
dominant clustering of the ${K^-}pp$ system in the configuration $\Lambda
(1405)+p$ allows us to calculate the binding energy to good accuracy within a
simple cluster approach for the differential Faddeev equations. The theoretical
discrepancies in the binding energy and width for the ${K^-}pp$ system related
to the different $NN$ and $\overset{\_}{K}N$ interactions are addressed.
