  The type-I region of phase transitions at finite temperature of the
U(1)-Higgs theory in 3+1 dimensions is investigated in detail using a Wilsonian
renormalisation group. We consider in particular the quantitative effects
induced through the cross-over of the scale-dependent Abelian charge from the
Gaussian to a non-trivial Abelian fixed point. As a result, the strength of the
first-order phase transition is weakened. Analytical solutions to approximate
flow equations are obtained, and all characteristics of the phase transition
are discussed and compared to the results obtained from perturbation theory. In
addition, we present a detailed quantitative study regarding the dependence of
the physical observables on the coarse-graining scheme. This results in
error-bars for the regularisation scheme (RS) dependence. We find quantitative
evidence for an intimate link between the RS dependence and truncations of flow
equations.
