  In this paper, we extend previous work on scalar $\phi^4$ theory using the
Source Galerkin method. This approach is based on finding solutions $Z[J]$ to
the lattice functional equations for field theories in the presence of an
external source $J$. Using polynomial expansions for the generating functional
$Z$, we calculate propagators and mass-gaps for a number of systems. These
calculations are straightforward to perform and are executed rapidly compared
to Monte Carlo. The bulk of the computation involves a single matrix inversion.
The use of polynomial expansions illustrates in a clear and simple way the
ideas of the Source Galerkin method. But at the same time, this choice has
serious limitations. Even after exploiting symmetries, the size of calculations
become prohibitive except for small systems. The calculations in this paper
were made on a workstation of modest power using a fourth order polynomial
expansion for lattices of size $8^2$,$4^3$,$2^4$ in $2D$, $3D$, and $4D$. In
addition, we present an alternative to the Galerkin procedure that results in
sparse matrices to invert.
