  Embedding the lattice gauge theory into a continuum theory allows to use the
continuum action as trial action in the variational calculation. Only
originally divergent graphs contribute. This leads to a very simple scheme
which makes it possible to write down explicit expressions for the plaquette
energy $E$ for U(1) in arbitrary space time dimension for the first three
orders of the expansion. For dimensions three and four one can even go up to
fourth order. This allows a rather thorough empirical investigation of the
convergence properties of the $\delta $-expansion, in particular near the phase
transition or the transition region, respectively. As already found in previous
work, the principle of minimal sensitivity can be only applied for $\beta $
above a certain value, because otherwise no extremum with respect to the
variational parameter exists. One can, however, extend the range of
applicability down to small $\beta $, by calculating instead of $E$ some power
$E^\kappa $, or by performing an appropriate Pad\'e transformation. We find
excellent agreement with the data for $\beta $ above the transition region for
the second and higher orders. Below the transition region the agreement is
rather poor in low orders, but quite impressive in fourth order. For SU(2) we
performed the calculation up to second order. The agreement with the data is
somewhat worse than in the abelian case.
