We derive asymptotic formulas for the solutions of the mixed boundary value problem for
the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical
rods. One of the ends of each rod is set into a hole in the plate and the other one is
supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole
remaining part of the boundary. Elements of the junction are assumed to have contrasting
properties so that the small parameter, i.e. the relative thickness,
appears in the differential equation, too, while the asymptotic structures crucially
depend on the contrastness ratio. Asymptotic error estimates are derived in anisotropic
weighted Sobolev norms.